%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET809+4 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n007.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:26:23 AM UTC 2025
% Result : Theorem 0.12s 0.37s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : SET809+4 : TPTP v9.0.0. Released v3.2.0.
% 0.07/0.12 % Command : run_E %s %d THM
% 0.12/0.32 % Computer : n007.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Jun 20 05:44:35 EDT 2025
% 0.12/0.33 % CPUTime :
% 0.12/0.37 % SZS status Theorem
% 0.12/0.37 % SZS output start Proof
% 0.12/0.37 tff(apply_type, type, (
% 0.12/0.37 apply: ( $i * $i * $i ) > $o)).
% 0.12/0.37 tff(member_predicate_type, type, (
% 0.12/0.37 member_predicate: $i)).
% 0.12/0.37 tff(member_type, type, (
% 0.12/0.37 member: ( $i * $i ) > $o)).
% 0.12/0.37 tff(tptp_fun_A_13_type, type, (
% 0.12/0.37 tptp_fun_A_13: $i)).
% 0.12/0.37 tff(strict_order_type, type, (
% 0.12/0.37 strict_order: ( $i * $i ) > $o)).
% 0.12/0.37 tff(tptp_fun_Z_10_type, type, (
% 0.12/0.37 tptp_fun_Z_10: ( $i * $i ) > $i)).
% 0.12/0.37 tff(tptp_fun_X_12_type, type, (
% 0.12/0.37 tptp_fun_X_12: ( $i * $i ) > $i)).
% 0.12/0.37 tff(tptp_fun_Y_11_type, type, (
% 0.12/0.37 tptp_fun_Y_11: ( $i * $i ) > $i)).
% 0.12/0.37 tff(tptp_fun_X_9_type, type, (
% 0.12/0.37 tptp_fun_X_9: ( $i * $i ) > $i)).
% 0.12/0.37 tff(tptp_fun_Y_8_type, type, (
% 0.12/0.37 tptp_fun_Y_8: ( $i * $i ) > $i)).
% 0.12/0.37 tff(least_type, type, (
% 0.12/0.37 least: ( $i * $i * $i ) > $o)).
% 0.12/0.37 tff(tptp_fun_Y_4_type, type, (
% 0.12/0.37 tptp_fun_Y_4: ( $i * $i ) > $i)).
% 0.12/0.37 tff(subset_type, type, (
% 0.12/0.37 subset: ( $i * $i ) > $o)).
% 0.12/0.37 tff(strict_well_order_type, type, (
% 0.12/0.37 strict_well_order: ( $i * $i ) > $o)).
% 0.12/0.37 tff(tptp_fun_A_5_type, type, (
% 0.12/0.37 tptp_fun_A_5: ( $i * $i ) > $i)).
% 0.12/0.37 tff(tptp_fun_X_6_type, type, (
% 0.12/0.37 tptp_fun_X_6: ( $i * $i ) > $i)).
% 0.12/0.37 tff(set_type, type, (
% 0.12/0.37 set: $i > $o)).
% 0.12/0.37 tff(on_type, type, (
% 0.12/0.37 on: $i)).
% 0.12/0.37 tff(tptp_fun_X_3_type, type, (
% 0.12/0.37 tptp_fun_X_3: $i > $i)).
% 0.12/0.37 tff(1,plain,
% 0.12/0.37 (^[R: $i, E: $i] : rewrite((~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))))) <=> (~((~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))) | (~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(2,plain,
% 0.12/0.37 (![R: $i, E: $i] : (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))) | (~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.37 tff(3,plain,
% 0.12/0.37 (^[R: $i, E: $i] : refl((~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))))) <=> (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))))))),
% 0.12/0.37 inference(bind,[status(th)],[])).
% 0.12/0.37 tff(4,plain,
% 0.12/0.37 (![R: $i, E: $i] : (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E))))))))),
% 0.12/0.37 inference(quant_intro,[status(thm)],[3])).
% 0.12/0.37 tff(5,plain,
% 0.12/0.37 (^[R: $i, E: $i] : rewrite((~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))))) <=> (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(6,plain,
% 0.12/0.38 (![R: $i, E: $i] : (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E))))))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[5])).
% 0.12/0.38 tff(7,plain,
% 0.12/0.38 (![R: $i, E: $i] : (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))))) <=> ![R: $i, E: $i] : (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E))))))))),
% 0.12/0.38 inference(transitivity,[status(thm)],[6, 4])).
% 0.12/0.38 tff(8,plain,
% 0.12/0.38 (^[R: $i, E: $i] : trans(monotonicity(rewrite(((~strict_order(R, E)) | (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) <=> ((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))), trans(monotonicity(rewrite((~((~(member(tptp_fun_X_9(E, R), E) & member(tptp_fun_Y_8(E, R), E))) | (~(apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R)) & apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R)))))) <=> (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R)))))), rewrite((~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R)) & apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R)))) | (~(member(tptp_fun_X_12(E, R), E) & member(tptp_fun_Y_11(E, R), E) & member(tptp_fun_Z_10(E, R), E))))) <=> (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E))))), ((strict_order(R, E) | (~((~(member(tptp_fun_X_9(E, R), E) & member(tptp_fun_Y_8(E, R), E))) | (~(apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R)) & apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R)))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R)) & apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R)))) | (~(member(tptp_fun_X_12(E, R), E) & member(tptp_fun_Y_11(E, R), E) & member(tptp_fun_Z_10(E, R), E)))))) <=> (strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E))))))), rewrite((strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E))))) <=> (strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))), ((strict_order(R, E) | (~((~(member(tptp_fun_X_9(E, R), E) & member(tptp_fun_Y_8(E, R), E))) | (~(apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R)) & apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R)))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R)) & apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R)))) | (~(member(tptp_fun_X_12(E, R), E) & member(tptp_fun_Y_11(E, R), E) & member(tptp_fun_Z_10(E, R), E)))))) <=> (strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E))))))), ((((~strict_order(R, E)) | (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (strict_order(R, E) | (~((~(member(tptp_fun_X_9(E, R), E) & member(tptp_fun_Y_8(E, R), E))) | (~(apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R)) & apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R)))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R)) & apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R)))) | (~(member(tptp_fun_X_12(E, R), E) & member(tptp_fun_Y_11(E, R), E) & member(tptp_fun_Z_10(E, R), E))))))) <=> (((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))) & (strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))))), rewrite((((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))) & (strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))) <=> (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E))))))))), ((((~strict_order(R, E)) | (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (strict_order(R, E) | (~((~(member(tptp_fun_X_9(E, R), E) & member(tptp_fun_Y_8(E, R), E))) | (~(apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R)) & apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R)))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R)) & apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R)))) | (~(member(tptp_fun_X_12(E, R), E) & member(tptp_fun_Y_11(E, R), E) & member(tptp_fun_Z_10(E, R), E))))))) <=> (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E))))))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(9,plain,
% 0.12/0.38 (![R: $i, E: $i] : (((~strict_order(R, E)) | (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (strict_order(R, E) | (~((~(member(tptp_fun_X_9(E, R), E) & member(tptp_fun_Y_8(E, R), E))) | (~(apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R)) & apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R)))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R)) & apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R)))) | (~(member(tptp_fun_X_12(E, R), E) & member(tptp_fun_Y_11(E, R), E) & member(tptp_fun_Z_10(E, R), E))))))) <=> ![R: $i, E: $i] : (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E))))))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[8])).
% 0.12/0.38 tff(10,plain,
% 0.12/0.38 (^[R: $i, E: $i] : rewrite((((~strict_order(R, E)) | (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (strict_order(R, E) | ((~((~(member(tptp_fun_X_9(E, R), E) & member(tptp_fun_Y_8(E, R), E))) | (~(apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R)) & apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R)))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R)) & apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R)))) | (~(member(tptp_fun_X_12(E, R), E) & member(tptp_fun_Y_11(E, R), E) & member(tptp_fun_Z_10(E, R), E)))))))) <=> (((~strict_order(R, E)) | (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (strict_order(R, E) | (~((~(member(tptp_fun_X_9(E, R), E) & member(tptp_fun_Y_8(E, R), E))) | (~(apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R)) & apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R)))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R)) & apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R)))) | (~(member(tptp_fun_X_12(E, R), E) & member(tptp_fun_Y_11(E, R), E) & member(tptp_fun_Z_10(E, R), E))))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(11,plain,
% 0.12/0.38 (![R: $i, E: $i] : (((~strict_order(R, E)) | (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (strict_order(R, E) | ((~((~(member(tptp_fun_X_9(E, R), E) & member(tptp_fun_Y_8(E, R), E))) | (~(apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R)) & apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R)))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R)) & apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R)))) | (~(member(tptp_fun_X_12(E, R), E) & member(tptp_fun_Y_11(E, R), E) & member(tptp_fun_Z_10(E, R), E)))))))) <=> ![R: $i, E: $i] : (((~strict_order(R, E)) | (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (strict_order(R, E) | (~((~(member(tptp_fun_X_9(E, R), E) & member(tptp_fun_Y_8(E, R), E))) | (~(apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R)) & apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R)))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R)) & apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R)))) | (~(member(tptp_fun_X_12(E, R), E) & member(tptp_fun_Y_11(E, R), E) & member(tptp_fun_Z_10(E, R), E)))))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[10])).
% 0.12/0.38 tff(12,plain,
% 0.12/0.38 (![R: $i, E: $i] : (strict_order(R, E) <=> (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) <=> ![R: $i, E: $i] : (strict_order(R, E) <=> (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.12/0.38 inference(rewrite,[status(thm)],[])).
% 0.12/0.38 tff(13,plain,
% 0.12/0.38 (^[R: $i, E: $i] : rewrite((strict_order(R, E) <=> (![X: $i, Y: $i] : ((member(X, E) & member(Y, E)) => (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (((member(X, E) & member(Y, E)) & member(Z, E)) => ((apply(R, X, Y) & apply(R, Y, Z)) => apply(R, X, Z))))) <=> (strict_order(R, E) <=> (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))))),
% 0.12/0.38 inference(bind,[status(th)],[])).
% 0.12/0.38 tff(14,plain,
% 0.12/0.38 (![R: $i, E: $i] : (strict_order(R, E) <=> (![X: $i, Y: $i] : ((member(X, E) & member(Y, E)) => (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (((member(X, E) & member(Y, E)) & member(Z, E)) => ((apply(R, X, Y) & apply(R, Y, Z)) => apply(R, X, Z))))) <=> ![R: $i, E: $i] : (strict_order(R, E) <=> (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.12/0.38 inference(quant_intro,[status(thm)],[13])).
% 0.12/0.38 tff(15,axiom,(![R: $i, E: $i] : (strict_order(R, E) <=> (![X: $i, Y: $i] : ((member(X, E) & member(Y, E)) => (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (((member(X, E) & member(Y, E)) & member(Z, E)) => ((apply(R, X, Y) & apply(R, Y, Z)) => apply(R, X, Z)))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+4.ax','strict_order')).
% 0.12/0.38 tff(16,plain,
% 0.12/0.38 (![R: $i, E: $i] : (strict_order(R, E) <=> (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[15, 14])).
% 0.12/0.38 tff(17,plain,
% 0.12/0.38 (![R: $i, E: $i] : (strict_order(R, E) <=> (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E))))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.12/0.38 tff(18,plain,(
% 0.12/0.38 ![R: $i, E: $i] : (((~strict_order(R, E)) | (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (strict_order(R, E) | ((~((~(member(tptp_fun_X_9(E, R), E) & member(tptp_fun_Y_8(E, R), E))) | (~(apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R)) & apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R)))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R)) & apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R)))) | (~(member(tptp_fun_X_12(E, R), E) & member(tptp_fun_Y_11(E, R), E) & member(tptp_fun_Z_10(E, R), E))))))))),
% 0.12/0.38 inference(skolemize,[status(sab)],[17])).
% 0.12/0.38 tff(19,plain,
% 0.12/0.38 (![R: $i, E: $i] : (((~strict_order(R, E)) | (![X: $i, Y: $i] : ((~(member(X, E) & member(Y, E))) | (~(apply(R, X, Y) & apply(R, Y, X)))) & ![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~(apply(R, X, Y) & apply(R, Y, Z))) | (~(member(X, E) & member(Y, E) & member(Z, E)))))) & (strict_order(R, E) | (~((~(member(tptp_fun_X_9(E, R), E) & member(tptp_fun_Y_8(E, R), E))) | (~(apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R)) & apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R)))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R)) & apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R)))) | (~(member(tptp_fun_X_12(E, R), E) & member(tptp_fun_Y_11(E, R), E) & member(tptp_fun_Z_10(E, R), E)))))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[18, 11])).
% 0.12/0.38 tff(20,plain,
% 0.12/0.38 (![R: $i, E: $i] : (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E))))))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[19, 9])).
% 0.12/0.38 tff(21,plain,
% 0.12/0.38 (![R: $i, E: $i] : (~((~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E)))))))) | (~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E))))))))),
% 0.12/0.38 inference(modus_ponens,[status(thm)],[20, 7])).
% 0.12/0.38 tff(22,plain,
% 0.12/0.38 (![R: $i, E: $i] : (~((~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))) | (~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[21, 2])).
% 0.12/0.39 tff(23,plain,
% 0.12/0.39 (((~![R: $i, E: $i] : (~((~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))) | (~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(strict_order(member_predicate, A!13) | (~((~member(tptp_fun_X_9(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_8(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_X_9(A!13, member_predicate), tptp_fun_Y_8(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_8(A!13, member_predicate), tptp_fun_X_9(A!13, member_predicate))))) | (~((~member(tptp_fun_Z_10(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_11(A!13, member_predicate), A!13)) | (~member(tptp_fun_X_12(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_Y_11(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Y_11(A!13, member_predicate))) | apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate)))))) | (~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13))))))))))) <=> ((~![R: $i, E: $i] : (~((~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))) | (~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(strict_order(member_predicate, A!13) | (~((~member(tptp_fun_X_9(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_8(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_X_9(A!13, member_predicate), tptp_fun_Y_8(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_8(A!13, member_predicate), tptp_fun_X_9(A!13, member_predicate))))) | (~((~member(tptp_fun_Z_10(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_11(A!13, member_predicate), A!13)) | (~member(tptp_fun_X_12(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_Y_11(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Y_11(A!13, member_predicate))) | apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate)))))) | (~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13)))))))))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(24,plain,
% 0.12/0.39 ((~((~(strict_order(member_predicate, A!13) | (~((~member(tptp_fun_X_9(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_8(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_X_9(A!13, member_predicate), tptp_fun_Y_8(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_8(A!13, member_predicate), tptp_fun_X_9(A!13, member_predicate))))) | (~(apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate)) | (~apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Y_11(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_11(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate))) | (~member(tptp_fun_X_12(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_11(A!13, member_predicate), A!13)) | (~member(tptp_fun_Z_10(A!13, member_predicate), A!13)))))) | (~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~member(X, A!13)) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13)) | (~member(Z, A!13)))))))))) <=> (~((~(strict_order(member_predicate, A!13) | (~((~member(tptp_fun_X_9(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_8(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_X_9(A!13, member_predicate), tptp_fun_Y_8(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_8(A!13, member_predicate), tptp_fun_X_9(A!13, member_predicate))))) | (~((~member(tptp_fun_Z_10(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_11(A!13, member_predicate), A!13)) | (~member(tptp_fun_X_12(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_Y_11(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Y_11(A!13, member_predicate))) | apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate)))))) | (~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13))))))))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(25,plain,
% 0.12/0.39 (((~![R: $i, E: $i] : (~((~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))) | (~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(strict_order(member_predicate, A!13) | (~((~member(tptp_fun_X_9(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_8(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_X_9(A!13, member_predicate), tptp_fun_Y_8(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_8(A!13, member_predicate), tptp_fun_X_9(A!13, member_predicate))))) | (~(apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate)) | (~apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Y_11(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_11(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate))) | (~member(tptp_fun_X_12(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_11(A!13, member_predicate), A!13)) | (~member(tptp_fun_Z_10(A!13, member_predicate), A!13)))))) | (~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~member(X, A!13)) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13)) | (~member(Z, A!13))))))))))) <=> ((~![R: $i, E: $i] : (~((~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))) | (~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(strict_order(member_predicate, A!13) | (~((~member(tptp_fun_X_9(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_8(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_X_9(A!13, member_predicate), tptp_fun_Y_8(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_8(A!13, member_predicate), tptp_fun_X_9(A!13, member_predicate))))) | (~((~member(tptp_fun_Z_10(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_11(A!13, member_predicate), A!13)) | (~member(tptp_fun_X_12(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_Y_11(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Y_11(A!13, member_predicate))) | apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate)))))) | (~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13)))))))))))),
% 0.12/0.39 inference(monotonicity,[status(thm)],[24])).
% 0.12/0.39 tff(26,plain,
% 0.12/0.39 (((~![R: $i, E: $i] : (~((~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))) | (~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(strict_order(member_predicate, A!13) | (~((~member(tptp_fun_X_9(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_8(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_X_9(A!13, member_predicate), tptp_fun_Y_8(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_8(A!13, member_predicate), tptp_fun_X_9(A!13, member_predicate))))) | (~(apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate)) | (~apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Y_11(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_11(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate))) | (~member(tptp_fun_X_12(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_11(A!13, member_predicate), A!13)) | (~member(tptp_fun_Z_10(A!13, member_predicate), A!13)))))) | (~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~member(X, A!13)) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13)) | (~member(Z, A!13))))))))))) <=> ((~![R: $i, E: $i] : (~((~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))) | (~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(strict_order(member_predicate, A!13) | (~((~member(tptp_fun_X_9(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_8(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_X_9(A!13, member_predicate), tptp_fun_Y_8(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_8(A!13, member_predicate), tptp_fun_X_9(A!13, member_predicate))))) | (~((~member(tptp_fun_Z_10(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_11(A!13, member_predicate), A!13)) | (~member(tptp_fun_X_12(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_Y_11(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Y_11(A!13, member_predicate))) | apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate)))))) | (~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13)))))))))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[25, 23])).
% 0.19/0.40 tff(27,plain,
% 0.19/0.40 ((~![R: $i, E: $i] : (~((~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))) | (~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(strict_order(member_predicate, A!13) | (~((~member(tptp_fun_X_9(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_8(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_X_9(A!13, member_predicate), tptp_fun_Y_8(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_8(A!13, member_predicate), tptp_fun_X_9(A!13, member_predicate))))) | (~(apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate)) | (~apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Y_11(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_11(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate))) | (~member(tptp_fun_X_12(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_11(A!13, member_predicate), A!13)) | (~member(tptp_fun_Z_10(A!13, member_predicate), A!13)))))) | (~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~member(X, A!13)) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13)) | (~member(Z, A!13))))))))))),
% 0.19/0.40 inference(quant_inst,[status(thm)],[])).
% 0.19/0.40 tff(28,plain,
% 0.19/0.40 ((~![R: $i, E: $i] : (~((~(strict_order(R, E) | (~((~member(tptp_fun_X_9(E, R), E)) | (~member(tptp_fun_Y_8(E, R), E)) | (~apply(R, tptp_fun_X_9(E, R), tptp_fun_Y_8(E, R))) | (~apply(R, tptp_fun_Y_8(E, R), tptp_fun_X_9(E, R))))) | (~(apply(R, tptp_fun_X_12(E, R), tptp_fun_Z_10(E, R)) | (~apply(R, tptp_fun_X_12(E, R), tptp_fun_Y_11(E, R))) | (~apply(R, tptp_fun_Y_11(E, R), tptp_fun_Z_10(E, R))) | (~member(tptp_fun_X_12(E, R), E)) | (~member(tptp_fun_Y_11(E, R), E)) | (~member(tptp_fun_Z_10(E, R), E)))))) | (~((~strict_order(R, E)) | (~((~![X: $i, Y: $i] : ((~member(Y, E)) | (~member(X, E)) | (~apply(R, X, Y)) | (~apply(R, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : (apply(R, X, Z) | (~apply(R, X, Y)) | (~apply(R, Y, Z)) | (~member(X, E)) | (~member(Y, E)) | (~member(Z, E))))))))))) | (~((~(strict_order(member_predicate, A!13) | (~((~member(tptp_fun_X_9(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_8(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_X_9(A!13, member_predicate), tptp_fun_Y_8(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_8(A!13, member_predicate), tptp_fun_X_9(A!13, member_predicate))))) | (~((~member(tptp_fun_Z_10(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_11(A!13, member_predicate), A!13)) | (~member(tptp_fun_X_12(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_Y_11(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Y_11(A!13, member_predicate))) | apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate)))))) | (~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13))))))))))),
% 0.19/0.40 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.19/0.40 tff(29,plain,
% 0.19/0.40 (~((~(strict_order(member_predicate, A!13) | (~((~member(tptp_fun_X_9(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_8(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_X_9(A!13, member_predicate), tptp_fun_Y_8(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_8(A!13, member_predicate), tptp_fun_X_9(A!13, member_predicate))))) | (~((~member(tptp_fun_Z_10(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_11(A!13, member_predicate), A!13)) | (~member(tptp_fun_X_12(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_Y_11(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Y_11(A!13, member_predicate))) | apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate)))))) | (~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13)))))))))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[28, 22])).
% 0.19/0.40 tff(30,plain,
% 0.19/0.40 (((~(strict_order(member_predicate, A!13) | (~((~member(tptp_fun_X_9(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_8(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_X_9(A!13, member_predicate), tptp_fun_Y_8(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_Y_8(A!13, member_predicate), tptp_fun_X_9(A!13, member_predicate))))) | (~((~member(tptp_fun_Z_10(A!13, member_predicate), A!13)) | (~member(tptp_fun_Y_11(A!13, member_predicate), A!13)) | (~member(tptp_fun_X_12(A!13, member_predicate), A!13)) | (~apply(member_predicate, tptp_fun_Y_11(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate))) | (~apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Y_11(A!13, member_predicate))) | apply(member_predicate, tptp_fun_X_12(A!13, member_predicate), tptp_fun_Z_10(A!13, member_predicate)))))) | (~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13))))))))) | ((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13)))))))),
% 0.19/0.40 inference(tautology,[status(thm)],[])).
% 0.19/0.40 tff(31,plain,
% 0.19/0.40 ((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13))))))),
% 0.19/0.40 inference(unit_resolution,[status(thm)],[30, 29])).
% 0.19/0.40 tff(32,plain,
% 0.19/0.40 (^[R: $i, E: $i] : rewrite((~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))))) <=> (~((~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))) | (~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | least(tptp_fun_Y_4(A, R), R, A) | ![X: $i] : (~member(X, A)))))))))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(33,plain,
% 0.19/0.40 (![R: $i, E: $i] : (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))))) <=> ![R: $i, E: $i] : (~((~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))) | (~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | least(tptp_fun_Y_4(A, R), R, A) | ![X: $i] : (~member(X, A))))))))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[32])).
% 0.19/0.40 tff(34,plain,
% 0.19/0.40 (^[R: $i, E: $i] : refl((~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))))) <=> (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(35,plain,
% 0.19/0.40 (![R: $i, E: $i] : (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))))) <=> ![R: $i, E: $i] : (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[34])).
% 0.19/0.40 tff(36,plain,
% 0.19/0.40 (^[R: $i, E: $i] : rewrite((~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))))) <=> (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))))))),
% 0.19/0.40 inference(bind,[status(th)],[])).
% 0.19/0.40 tff(37,plain,
% 0.19/0.40 (![R: $i, E: $i] : (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))))) <=> ![R: $i, E: $i] : (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))))))),
% 0.19/0.40 inference(quant_intro,[status(thm)],[36])).
% 0.19/0.40 tff(38,plain,
% 0.19/0.40 (![R: $i, E: $i] : (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))))) <=> ![R: $i, E: $i] : (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))))))),
% 0.19/0.40 inference(transitivity,[status(thm)],[37, 35])).
% 0.19/0.40 tff(39,plain,
% 0.19/0.40 (^[R: $i, E: $i] : trans(monotonicity(rewrite(((~strict_well_order(R, E)) | (strict_order(R, E) & ![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A)))) <=> ((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))), trans(monotonicity(rewrite((subset(tptp_fun_A_5(E, R), E) & member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R)) & ![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))) <=> (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))), ((strict_well_order(R, E) | (~strict_order(R, E)) | (subset(tptp_fun_A_5(E, R), E) & member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R)) & ![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))) <=> (strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))))), rewrite((strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))) <=> (strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))), ((strict_well_order(R, E) | (~strict_order(R, E)) | (subset(tptp_fun_A_5(E, R), E) & member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R)) & ![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))) <=> (strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))))), ((((~strict_well_order(R, E)) | (strict_order(R, E) & ![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A)))) & (strict_well_order(R, E) | (~strict_order(R, E)) | (subset(tptp_fun_A_5(E, R), E) & member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R)) & ![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))) <=> (((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A)))))) & (strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))))), rewrite((((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A)))))) & (strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))) <=> (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))))))), ((((~strict_well_order(R, E)) | (strict_order(R, E) & ![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A)))) & (strict_well_order(R, E) | (~strict_order(R, E)) | (subset(tptp_fun_A_5(E, R), E) & member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R)) & ![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))) <=> (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(40,plain,
% 0.19/0.41 (![R: $i, E: $i] : (((~strict_well_order(R, E)) | (strict_order(R, E) & ![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A)))) & (strict_well_order(R, E) | (~strict_order(R, E)) | (subset(tptp_fun_A_5(E, R), E) & member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R)) & ![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))) <=> ![R: $i, E: $i] : (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[39])).
% 0.19/0.41 tff(41,plain,
% 0.19/0.41 (^[R: $i, E: $i] : rewrite((((~strict_well_order(R, E)) | (strict_order(R, E) & ![A: $i] : (((~subset(A, E)) | ![X: $i] : (~member(X, A))) | least(tptp_fun_Y_4(A, R), R, A)))) & (strict_well_order(R, E) | ((~strict_order(R, E)) | ((subset(tptp_fun_A_5(E, R), E) & member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) & ![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))) <=> (((~strict_well_order(R, E)) | (strict_order(R, E) & ![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A)))) & (strict_well_order(R, E) | (~strict_order(R, E)) | (subset(tptp_fun_A_5(E, R), E) & member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R)) & ![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(42,plain,
% 0.19/0.41 (![R: $i, E: $i] : (((~strict_well_order(R, E)) | (strict_order(R, E) & ![A: $i] : (((~subset(A, E)) | ![X: $i] : (~member(X, A))) | least(tptp_fun_Y_4(A, R), R, A)))) & (strict_well_order(R, E) | ((~strict_order(R, E)) | ((subset(tptp_fun_A_5(E, R), E) & member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) & ![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))) <=> ![R: $i, E: $i] : (((~strict_well_order(R, E)) | (strict_order(R, E) & ![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A)))) & (strict_well_order(R, E) | (~strict_order(R, E)) | (subset(tptp_fun_A_5(E, R), E) & member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R)) & ![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[41])).
% 0.19/0.41 tff(43,plain,
% 0.19/0.41 (![R: $i, E: $i] : (strict_well_order(R, E) <=> (strict_order(R, E) & ![A: $i] : ((~(subset(A, E) & ?[X: $i] : member(X, A))) | ?[Y: $i] : least(Y, R, A)))) <=> ![R: $i, E: $i] : (strict_well_order(R, E) <=> (strict_order(R, E) & ![A: $i] : ((~(subset(A, E) & ?[X: $i] : member(X, A))) | ?[Y: $i] : least(Y, R, A))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(44,plain,
% 0.19/0.41 (^[R: $i, E: $i] : rewrite((strict_well_order(R, E) <=> (strict_order(R, E) & ![A: $i] : ((subset(A, E) & ?[X: $i] : member(X, A)) => ?[Y: $i] : least(Y, R, A)))) <=> (strict_well_order(R, E) <=> (strict_order(R, E) & ![A: $i] : ((~(subset(A, E) & ?[X: $i] : member(X, A))) | ?[Y: $i] : least(Y, R, A)))))),
% 0.19/0.41 inference(bind,[status(th)],[])).
% 0.19/0.41 tff(45,plain,
% 0.19/0.41 (![R: $i, E: $i] : (strict_well_order(R, E) <=> (strict_order(R, E) & ![A: $i] : ((subset(A, E) & ?[X: $i] : member(X, A)) => ?[Y: $i] : least(Y, R, A)))) <=> ![R: $i, E: $i] : (strict_well_order(R, E) <=> (strict_order(R, E) & ![A: $i] : ((~(subset(A, E) & ?[X: $i] : member(X, A))) | ?[Y: $i] : least(Y, R, A))))),
% 0.19/0.41 inference(quant_intro,[status(thm)],[44])).
% 0.19/0.41 tff(46,axiom,(![R: $i, E: $i] : (strict_well_order(R, E) <=> (strict_order(R, E) & ![A: $i] : ((subset(A, E) & ?[X: $i] : member(X, A)) => ?[Y: $i] : least(Y, R, A))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+4.ax','strict_well_order')).
% 0.19/0.41 tff(47,plain,
% 0.19/0.41 (![R: $i, E: $i] : (strict_well_order(R, E) <=> (strict_order(R, E) & ![A: $i] : ((~(subset(A, E) & ?[X: $i] : member(X, A))) | ?[Y: $i] : least(Y, R, A))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.19/0.41 tff(48,plain,
% 0.19/0.41 (![R: $i, E: $i] : (strict_well_order(R, E) <=> (strict_order(R, E) & ![A: $i] : ((~(subset(A, E) & ?[X: $i] : member(X, A))) | ?[Y: $i] : least(Y, R, A))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[47, 43])).
% 0.19/0.41 tff(49,plain,(
% 0.19/0.41 ![R: $i, E: $i] : (((~strict_well_order(R, E)) | (strict_order(R, E) & ![A: $i] : (((~subset(A, E)) | ![X: $i] : (~member(X, A))) | least(tptp_fun_Y_4(A, R), R, A)))) & (strict_well_order(R, E) | ((~strict_order(R, E)) | ((subset(tptp_fun_A_5(E, R), E) & member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) & ![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))),
% 0.19/0.41 inference(skolemize,[status(sab)],[48])).
% 0.19/0.41 tff(50,plain,
% 0.19/0.41 (![R: $i, E: $i] : (((~strict_well_order(R, E)) | (strict_order(R, E) & ![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A)))) & (strict_well_order(R, E) | (~strict_order(R, E)) | (subset(tptp_fun_A_5(E, R), E) & member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R)) & ![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[49, 42])).
% 0.19/0.41 tff(51,plain,
% 0.19/0.41 (![R: $i, E: $i] : (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[50, 40])).
% 0.19/0.41 tff(52,plain,
% 0.19/0.41 (![R: $i, E: $i] : (~((~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | ![X: $i] : (~member(X, A)) | least(tptp_fun_Y_4(A, R), R, A))))))) | (~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R))))))))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[51, 38])).
% 0.19/0.41 tff(53,plain,
% 0.19/0.41 (![R: $i, E: $i] : (~((~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))) | (~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | least(tptp_fun_Y_4(A, R), R, A) | ![X: $i] : (~member(X, A))))))))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[52, 33])).
% 0.19/0.41 tff(54,plain,
% 0.19/0.41 (((~![R: $i, E: $i] : (~((~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))) | (~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | least(tptp_fun_Y_4(A, R), R, A) | ![X: $i] : (~member(X, A))))))))))) | (~((~(strict_well_order(member_predicate, A!13) | (~strict_order(member_predicate, A!13)) | (~((~![Y: $i] : (~least(Y, member_predicate, tptp_fun_A_5(A!13, member_predicate)))) | (~member(tptp_fun_X_6(A!13, member_predicate), tptp_fun_A_5(A!13, member_predicate))) | (~subset(tptp_fun_A_5(A!13, member_predicate), A!13)))))) | (~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A))))))))))) <=> ((~![R: $i, E: $i] : (~((~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))) | (~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | least(tptp_fun_Y_4(A, R), R, A) | ![X: $i] : (~member(X, A))))))))))) | (~((~(strict_well_order(member_predicate, A!13) | (~strict_order(member_predicate, A!13)) | (~((~![Y: $i] : (~least(Y, member_predicate, tptp_fun_A_5(A!13, member_predicate)))) | (~member(tptp_fun_X_6(A!13, member_predicate), tptp_fun_A_5(A!13, member_predicate))) | (~subset(tptp_fun_A_5(A!13, member_predicate), A!13)))))) | (~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A)))))))))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(55,plain,
% 0.19/0.41 ((~((~(strict_well_order(member_predicate, A!13) | (~strict_order(member_predicate, A!13)) | (~((~subset(tptp_fun_A_5(A!13, member_predicate), A!13)) | (~member(tptp_fun_X_6(A!13, member_predicate), tptp_fun_A_5(A!13, member_predicate))) | (~![Y: $i] : (~least(Y, member_predicate, tptp_fun_A_5(A!13, member_predicate)))))))) | (~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A)))))))))) <=> (~((~(strict_well_order(member_predicate, A!13) | (~strict_order(member_predicate, A!13)) | (~((~![Y: $i] : (~least(Y, member_predicate, tptp_fun_A_5(A!13, member_predicate)))) | (~member(tptp_fun_X_6(A!13, member_predicate), tptp_fun_A_5(A!13, member_predicate))) | (~subset(tptp_fun_A_5(A!13, member_predicate), A!13)))))) | (~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A))))))))))),
% 0.19/0.41 inference(rewrite,[status(thm)],[])).
% 0.19/0.41 tff(56,plain,
% 0.19/0.41 (((~![R: $i, E: $i] : (~((~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))) | (~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | least(tptp_fun_Y_4(A, R), R, A) | ![X: $i] : (~member(X, A))))))))))) | (~((~(strict_well_order(member_predicate, A!13) | (~strict_order(member_predicate, A!13)) | (~((~subset(tptp_fun_A_5(A!13, member_predicate), A!13)) | (~member(tptp_fun_X_6(A!13, member_predicate), tptp_fun_A_5(A!13, member_predicate))) | (~![Y: $i] : (~least(Y, member_predicate, tptp_fun_A_5(A!13, member_predicate)))))))) | (~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A))))))))))) <=> ((~![R: $i, E: $i] : (~((~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))) | (~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | least(tptp_fun_Y_4(A, R), R, A) | ![X: $i] : (~member(X, A))))))))))) | (~((~(strict_well_order(member_predicate, A!13) | (~strict_order(member_predicate, A!13)) | (~((~![Y: $i] : (~least(Y, member_predicate, tptp_fun_A_5(A!13, member_predicate)))) | (~member(tptp_fun_X_6(A!13, member_predicate), tptp_fun_A_5(A!13, member_predicate))) | (~subset(tptp_fun_A_5(A!13, member_predicate), A!13)))))) | (~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A)))))))))))),
% 0.19/0.41 inference(monotonicity,[status(thm)],[55])).
% 0.19/0.41 tff(57,plain,
% 0.19/0.41 (((~![R: $i, E: $i] : (~((~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))) | (~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | least(tptp_fun_Y_4(A, R), R, A) | ![X: $i] : (~member(X, A))))))))))) | (~((~(strict_well_order(member_predicate, A!13) | (~strict_order(member_predicate, A!13)) | (~((~subset(tptp_fun_A_5(A!13, member_predicate), A!13)) | (~member(tptp_fun_X_6(A!13, member_predicate), tptp_fun_A_5(A!13, member_predicate))) | (~![Y: $i] : (~least(Y, member_predicate, tptp_fun_A_5(A!13, member_predicate)))))))) | (~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A))))))))))) <=> ((~![R: $i, E: $i] : (~((~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))) | (~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | least(tptp_fun_Y_4(A, R), R, A) | ![X: $i] : (~member(X, A))))))))))) | (~((~(strict_well_order(member_predicate, A!13) | (~strict_order(member_predicate, A!13)) | (~((~![Y: $i] : (~least(Y, member_predicate, tptp_fun_A_5(A!13, member_predicate)))) | (~member(tptp_fun_X_6(A!13, member_predicate), tptp_fun_A_5(A!13, member_predicate))) | (~subset(tptp_fun_A_5(A!13, member_predicate), A!13)))))) | (~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A)))))))))))),
% 0.19/0.41 inference(transitivity,[status(thm)],[56, 54])).
% 0.19/0.41 tff(58,plain,
% 0.19/0.41 ((~![R: $i, E: $i] : (~((~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))) | (~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | least(tptp_fun_Y_4(A, R), R, A) | ![X: $i] : (~member(X, A))))))))))) | (~((~(strict_well_order(member_predicate, A!13) | (~strict_order(member_predicate, A!13)) | (~((~subset(tptp_fun_A_5(A!13, member_predicate), A!13)) | (~member(tptp_fun_X_6(A!13, member_predicate), tptp_fun_A_5(A!13, member_predicate))) | (~![Y: $i] : (~least(Y, member_predicate, tptp_fun_A_5(A!13, member_predicate)))))))) | (~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A))))))))))),
% 0.19/0.41 inference(quant_inst,[status(thm)],[])).
% 0.19/0.41 tff(59,plain,
% 0.19/0.41 ((~![R: $i, E: $i] : (~((~(strict_well_order(R, E) | (~strict_order(R, E)) | (~((~subset(tptp_fun_A_5(E, R), E)) | (~member(tptp_fun_X_6(E, R), tptp_fun_A_5(E, R))) | (~![Y: $i] : (~least(Y, R, tptp_fun_A_5(E, R)))))))) | (~((~strict_well_order(R, E)) | (~((~strict_order(R, E)) | (~![A: $i] : ((~subset(A, E)) | least(tptp_fun_Y_4(A, R), R, A) | ![X: $i] : (~member(X, A))))))))))) | (~((~(strict_well_order(member_predicate, A!13) | (~strict_order(member_predicate, A!13)) | (~((~![Y: $i] : (~least(Y, member_predicate, tptp_fun_A_5(A!13, member_predicate)))) | (~member(tptp_fun_X_6(A!13, member_predicate), tptp_fun_A_5(A!13, member_predicate))) | (~subset(tptp_fun_A_5(A!13, member_predicate), A!13)))))) | (~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A))))))))))),
% 0.19/0.41 inference(modus_ponens,[status(thm)],[58, 57])).
% 0.19/0.41 tff(60,plain,
% 0.19/0.41 (~((~(strict_well_order(member_predicate, A!13) | (~strict_order(member_predicate, A!13)) | (~((~![Y: $i] : (~least(Y, member_predicate, tptp_fun_A_5(A!13, member_predicate)))) | (~member(tptp_fun_X_6(A!13, member_predicate), tptp_fun_A_5(A!13, member_predicate))) | (~subset(tptp_fun_A_5(A!13, member_predicate), A!13)))))) | (~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A)))))))))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[59, 53])).
% 0.19/0.42 tff(61,plain,
% 0.19/0.42 (((~(strict_well_order(member_predicate, A!13) | (~strict_order(member_predicate, A!13)) | (~((~![Y: $i] : (~least(Y, member_predicate, tptp_fun_A_5(A!13, member_predicate)))) | (~member(tptp_fun_X_6(A!13, member_predicate), tptp_fun_A_5(A!13, member_predicate))) | (~subset(tptp_fun_A_5(A!13, member_predicate), A!13)))))) | (~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A))))))))) | ((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A)))))))),
% 0.19/0.42 inference(tautology,[status(thm)],[])).
% 0.19/0.42 tff(62,plain,
% 0.19/0.42 ((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A))))))),
% 0.19/0.42 inference(unit_resolution,[status(thm)],[61, 60])).
% 0.19/0.42 tff(63,plain,
% 0.19/0.42 (^[A: $i] : rewrite((~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(64,plain,
% 0.19/0.42 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> ![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[63])).
% 0.19/0.42 tff(65,plain,
% 0.19/0.42 (^[A: $i] : refl((~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(66,plain,
% 0.19/0.42 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> ![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[65])).
% 0.19/0.42 tff(67,plain,
% 0.19/0.42 (^[A: $i] : rewrite((~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(68,plain,
% 0.19/0.42 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> ![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[67])).
% 0.19/0.42 tff(69,plain,
% 0.19/0.42 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))) <=> ![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.19/0.42 inference(transitivity,[status(thm)],[68, 66])).
% 0.19/0.42 tff(70,plain,
% 0.19/0.42 (^[A: $i] : trans(monotonicity(rewrite(((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) <=> ((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))), rewrite((member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))) <=> (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))), ((((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) <=> (((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))), rewrite((((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) <=> (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))), ((((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) <=> (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(71,plain,
% 0.19/0.42 (![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) <=> ![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[70])).
% 0.19/0.42 tff(72,plain,
% 0.19/0.42 (^[A: $i] : rewrite((((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | ((~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))) <=> (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(73,plain,
% 0.19/0.42 (![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | ((~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))) <=> ![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[72])).
% 0.19/0.42 tff(74,plain,
% 0.19/0.42 (![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) <=> ![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(75,plain,
% 0.19/0.42 (^[A: $i] : rewrite((member(A, on) <=> ((set(A) & strict_well_order(member_predicate, A)) & ![X: $i] : (member(X, A) => subset(X, A)))) <=> (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))))),
% 0.19/0.42 inference(bind,[status(th)],[])).
% 0.19/0.42 tff(76,plain,
% 0.19/0.42 (![A: $i] : (member(A, on) <=> ((set(A) & strict_well_order(member_predicate, A)) & ![X: $i] : (member(X, A) => subset(X, A)))) <=> ![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A))))),
% 0.19/0.42 inference(quant_intro,[status(thm)],[75])).
% 0.19/0.42 tff(77,axiom,(![A: $i] : (member(A, on) <=> ((set(A) & strict_well_order(member_predicate, A)) & ![X: $i] : (member(X, A) => subset(X, A))))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+4.ax','ordinal_number')).
% 0.19/0.42 tff(78,plain,
% 0.19/0.42 (![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[77, 76])).
% 0.19/0.42 tff(79,plain,
% 0.19/0.42 (![A: $i] : (member(A, on) <=> (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[78, 74])).
% 0.19/0.42 tff(80,plain,(
% 0.19/0.42 ![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | ((~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))))),
% 0.19/0.42 inference(skolemize,[status(sab)],[79])).
% 0.19/0.42 tff(81,plain,
% 0.19/0.42 (![A: $i] : (((~member(A, on)) | (set(A) & strict_well_order(member_predicate, A) & ![X: $i] : ((~member(X, A)) | subset(X, A)))) & (member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[80, 73])).
% 0.19/0.42 tff(82,plain,
% 0.19/0.42 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[81, 71])).
% 0.19/0.42 tff(83,plain,
% 0.19/0.42 (![A: $i] : (~((~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A))))))) | (~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A)))))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[82, 69])).
% 0.19/0.42 tff(84,plain,
% 0.19/0.42 (![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))),
% 0.19/0.42 inference(modus_ponens,[status(thm)],[83, 64])).
% 0.19/0.42 tff(85,plain,
% 0.19/0.42 (((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))))) <=> ((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(86,plain,
% 0.19/0.42 ((~((~(member(A!13, on) | (~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))))) <=> (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))))),
% 0.19/0.42 inference(rewrite,[status(thm)],[])).
% 0.19/0.42 tff(87,plain,
% 0.19/0.42 (((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))))) <=> ((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))))))),
% 0.19/0.43 inference(monotonicity,[status(thm)],[86])).
% 0.19/0.43 tff(88,plain,
% 0.19/0.43 (((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))))) <=> ((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))))))),
% 0.19/0.43 inference(transitivity,[status(thm)],[87, 85])).
% 0.19/0.43 tff(89,plain,
% 0.19/0.43 ((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))))),
% 0.19/0.43 inference(quant_inst,[status(thm)],[])).
% 0.19/0.43 tff(90,plain,
% 0.19/0.43 ((~![A: $i] : (~((~(member(A, on) | (~set(A)) | (~strict_well_order(member_predicate, A)) | (~((~member(tptp_fun_X_3(A), A)) | subset(tptp_fun_X_3(A), A))))) | (~((~member(A, on)) | (~((~set(A)) | (~strict_well_order(member_predicate, A)) | (~![X: $i] : ((~member(X, A)) | subset(X, A)))))))))) | (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[89, 88])).
% 0.19/0.43 tff(91,plain,
% 0.19/0.43 (~((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[90, 84])).
% 0.19/0.43 tff(92,plain,
% 0.19/0.43 (((~(member(A!13, on) | (~strict_well_order(member_predicate, A!13)) | (~set(A!13)) | (~((~member(tptp_fun_X_3(A!13), A!13)) | subset(tptp_fun_X_3(A!13), A!13))))) | (~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))))) | ((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))),
% 0.19/0.43 inference(tautology,[status(thm)],[])).
% 0.19/0.43 tff(93,plain,
% 0.19/0.43 ((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[92, 91])).
% 0.19/0.43 tff(94,plain,
% 0.19/0.43 ((~((~member(A!13, on)) | (~member(A!13, A!13)))) <=> (~((~member(A!13, on)) | (~member(A!13, A!13))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(95,plain,
% 0.19/0.43 ((~![A: $i] : ((~member(A, on)) | (~member(A, A)))) <=> (~![A: $i] : ((~member(A, on)) | (~member(A, A))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(96,plain,
% 0.19/0.43 ((~![A: $i] : (member(A, on) => (~member(A, A)))) <=> (~![A: $i] : ((~member(A, on)) | (~member(A, A))))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(97,axiom,(~![A: $i] : (member(A, on) => (~member(A, A)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','thV2')).
% 0.19/0.43 tff(98,plain,
% 0.19/0.43 (~![A: $i] : ((~member(A, on)) | (~member(A, A)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[97, 96])).
% 0.19/0.43 tff(99,plain,
% 0.19/0.43 (~![A: $i] : ((~member(A, on)) | (~member(A, A)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[98, 95])).
% 0.19/0.43 tff(100,plain,
% 0.19/0.43 (~![A: $i] : ((~member(A, on)) | (~member(A, A)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[99, 95])).
% 0.19/0.43 tff(101,plain,
% 0.19/0.43 (~![A: $i] : ((~member(A, on)) | (~member(A, A)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[100, 95])).
% 0.19/0.43 tff(102,plain,(
% 0.19/0.43 ~((~member(A!13, on)) | (~member(A!13, A!13)))),
% 0.19/0.43 inference(skolemize,[status(sab)],[101])).
% 0.19/0.43 tff(103,plain,
% 0.19/0.43 (~((~member(A!13, on)) | (~member(A!13, A!13)))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[102, 94])).
% 0.19/0.43 tff(104,plain,
% 0.19/0.43 (member(A!13, on)),
% 0.19/0.43 inference(or_elim,[status(thm)],[103])).
% 0.19/0.43 tff(105,plain,
% 0.19/0.43 ((~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))) | (~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))),
% 0.19/0.43 inference(tautology,[status(thm)],[])).
% 0.19/0.43 tff(106,plain,
% 0.19/0.43 ((~((~member(A!13, on)) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))))) | (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[105, 104])).
% 0.19/0.43 tff(107,plain,
% 0.19/0.43 (~((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13))))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[106, 93])).
% 0.19/0.43 tff(108,plain,
% 0.19/0.43 (((~set(A!13)) | (~strict_well_order(member_predicate, A!13)) | (~![X: $i] : ((~member(X, A!13)) | subset(X, A!13)))) | strict_well_order(member_predicate, A!13)),
% 0.19/0.43 inference(tautology,[status(thm)],[])).
% 0.19/0.43 tff(109,plain,
% 0.19/0.43 (strict_well_order(member_predicate, A!13)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[108, 107])).
% 0.19/0.43 tff(110,plain,
% 0.19/0.43 ((~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A)))))))) | (~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A))))))),
% 0.19/0.43 inference(tautology,[status(thm)],[])).
% 0.19/0.43 tff(111,plain,
% 0.19/0.43 ((~((~strict_well_order(member_predicate, A!13)) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A)))))))) | (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A))))))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[110, 109])).
% 0.19/0.43 tff(112,plain,
% 0.19/0.43 (~((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A)))))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[111, 62])).
% 0.19/0.43 tff(113,plain,
% 0.19/0.43 (((~strict_order(member_predicate, A!13)) | (~![A: $i] : ((~subset(A, A!13)) | least(tptp_fun_Y_4(A, member_predicate), member_predicate, A) | ![X: $i] : (~member(X, A))))) | strict_order(member_predicate, A!13)),
% 0.19/0.43 inference(tautology,[status(thm)],[])).
% 0.19/0.43 tff(114,plain,
% 0.19/0.43 (strict_order(member_predicate, A!13)),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[113, 112])).
% 0.19/0.43 tff(115,plain,
% 0.19/0.43 ((~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13)))))))) | (~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13))))))),
% 0.19/0.43 inference(tautology,[status(thm)],[])).
% 0.19/0.43 tff(116,plain,
% 0.19/0.43 ((~((~strict_order(member_predicate, A!13)) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13)))))))) | (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13))))))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[115, 114])).
% 0.19/0.43 tff(117,plain,
% 0.19/0.43 (~((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13)))))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[116, 31])).
% 0.19/0.43 tff(118,plain,
% 0.19/0.43 (((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~![X: $i, Y: $i, Z: $i] : ((~member(Z, A!13)) | apply(member_predicate, X, Z) | (~apply(member_predicate, X, Y)) | (~apply(member_predicate, Y, Z)) | (~member(X, A!13)) | (~member(Y, A!13))))) | ![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))),
% 0.19/0.43 inference(tautology,[status(thm)],[])).
% 0.19/0.43 tff(119,plain,
% 0.19/0.43 (![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))),
% 0.19/0.43 inference(unit_resolution,[status(thm)],[118, 117])).
% 0.19/0.43 tff(120,plain,
% 0.19/0.43 (^[X: $i, Y: $i] : refl((apply(member_predicate, X, Y) <=> member(X, Y)) <=> (apply(member_predicate, X, Y) <=> member(X, Y)))),
% 0.19/0.43 inference(bind,[status(th)],[])).
% 0.19/0.43 tff(121,plain,
% 0.19/0.43 (![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y)) <=> ![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))),
% 0.19/0.43 inference(quant_intro,[status(thm)],[120])).
% 0.19/0.43 tff(122,plain,
% 0.19/0.43 (![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y)) <=> ![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))),
% 0.19/0.43 inference(rewrite,[status(thm)],[])).
% 0.19/0.43 tff(123,axiom,(![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET006+4.ax','rel_member')).
% 0.19/0.43 tff(124,plain,
% 0.19/0.43 (![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[123, 122])).
% 0.19/0.43 tff(125,plain,(
% 0.19/0.43 ![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))),
% 0.19/0.43 inference(skolemize,[status(sab)],[124])).
% 0.19/0.43 tff(126,plain,
% 0.19/0.43 (![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))),
% 0.19/0.43 inference(modus_ponens,[status(thm)],[125, 121])).
% 0.19/0.43 tff(127,plain,
% 0.19/0.44 ((~![X: $i, Y: $i] : (apply(member_predicate, X, Y) <=> member(X, Y))) | (apply(member_predicate, A!13, A!13) <=> member(A!13, A!13))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(128,plain,
% 0.19/0.44 (apply(member_predicate, A!13, A!13) <=> member(A!13, A!13)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[127, 126])).
% 0.19/0.44 tff(129,plain,
% 0.19/0.44 (member(A!13, A!13)),
% 0.19/0.44 inference(or_elim,[status(thm)],[103])).
% 0.19/0.44 tff(130,plain,
% 0.19/0.44 ((~(apply(member_predicate, A!13, A!13) <=> member(A!13, A!13))) | apply(member_predicate, A!13, A!13) | (~member(A!13, A!13))),
% 0.19/0.44 inference(tautology,[status(thm)],[])).
% 0.19/0.44 tff(131,plain,
% 0.19/0.44 ((~(apply(member_predicate, A!13, A!13) <=> member(A!13, A!13))) | apply(member_predicate, A!13, A!13)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[130, 129])).
% 0.19/0.44 tff(132,plain,
% 0.19/0.44 (apply(member_predicate, A!13, A!13)),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[131, 128])).
% 0.19/0.44 tff(133,plain,
% 0.19/0.44 (((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | ((~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13)))) <=> ((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(134,plain,
% 0.19/0.44 (((~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13)) | (~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13))) <=> ((~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13)))),
% 0.19/0.44 inference(rewrite,[status(thm)],[])).
% 0.19/0.44 tff(135,plain,
% 0.19/0.44 (((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | ((~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13)) | (~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13)))) <=> ((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | ((~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13))))),
% 0.19/0.44 inference(monotonicity,[status(thm)],[134])).
% 0.19/0.44 tff(136,plain,
% 0.19/0.44 (((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | ((~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13)) | (~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13)))) <=> ((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13)))),
% 0.19/0.44 inference(transitivity,[status(thm)],[135, 133])).
% 0.19/0.44 tff(137,plain,
% 0.19/0.44 ((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | ((~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13)) | (~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13)))),
% 0.19/0.44 inference(quant_inst,[status(thm)],[])).
% 0.19/0.44 tff(138,plain,
% 0.19/0.44 ((~![X: $i, Y: $i] : ((~member(Y, A!13)) | (~apply(member_predicate, X, Y)) | (~member(X, A!13)) | (~apply(member_predicate, Y, X)))) | (~member(A!13, A!13)) | (~apply(member_predicate, A!13, A!13))),
% 0.19/0.44 inference(modus_ponens,[status(thm)],[137, 136])).
% 0.19/0.44 tff(139,plain,
% 0.19/0.44 ($false),
% 0.19/0.44 inference(unit_resolution,[status(thm)],[138, 129, 132, 119])).
% 0.19/0.44 % SZS output end Proof
% 0.19/0.44 % E exiting
%------------------------------------------------------------------------------