%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET020+1 : TPTP v9.0.0. Bugfixed v5.4.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n001.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 05:24:51 AM UTC 2025
% Result : Theorem 0.21s 0.40s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.13 % Problem : SET020+1 : TPTP v9.0.0. Bugfixed v5.4.0.
% 0.04/0.13 % Command : run_E %s %d THM
% 0.14/0.35 % Computer : n001.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Jun 20 04:05:49 EDT 2025
% 0.14/0.35 % CPUTime :
% 0.21/0.40 % SZS status Theorem
% 0.21/0.40 % SZS output start Proof
% 0.21/0.40 tff(tptp_fun_U_8_type, type, (
% 0.21/0.40 tptp_fun_U_8: $i)).
% 0.21/0.40 tff(first_type, type, (
% 0.21/0.40 first: $i > $i)).
% 0.21/0.40 tff(ordered_pair_type, type, (
% 0.21/0.40 ordered_pair: ( $i * $i ) > $i)).
% 0.21/0.40 tff(tptp_fun_V_7_type, type, (
% 0.21/0.40 tptp_fun_V_7: $i)).
% 0.21/0.40 tff(second_type, type, (
% 0.21/0.40 second: $i > $i)).
% 0.21/0.40 tff(member_type, type, (
% 0.21/0.40 member: ( $i * $i ) > $o)).
% 0.21/0.40 tff(universal_class_type, type, (
% 0.21/0.40 universal_class: $i)).
% 0.21/0.40 tff(1,assumption,((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7))), introduced(assumption)).
% 0.21/0.40 tff(2,plain,
% 0.21/0.40 ((~(((first(ordered_pair(U!8, V!7)) = U!8) & (second(ordered_pair(U!8, V!7)) = V!7)) | (~member(V!7, universal_class)) | (~member(U!8, universal_class)))) <=> (~(((first(ordered_pair(U!8, V!7)) = U!8) & (second(ordered_pair(U!8, V!7)) = V!7)) | (~member(V!7, universal_class)) | (~member(U!8, universal_class))))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(3,plain,
% 0.21/0.40 ((~![U: $i, V: $i] : (((first(ordered_pair(U, V)) = U) & (second(ordered_pair(U, V)) = V)) | (~member(V, universal_class)) | (~member(U, universal_class)))) <=> (~![U: $i, V: $i] : (((first(ordered_pair(U, V)) = U) & (second(ordered_pair(U, V)) = V)) | (~member(V, universal_class)) | (~member(U, universal_class))))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(4,plain,
% 0.21/0.40 ((~![U: $i, V: $i, X: $i] : (((member(U, universal_class) & member(V, universal_class)) & (X = ordered_pair(U, V))) => ((first(X) = U) & (second(X) = V)))) <=> (~![U: $i, V: $i] : (((first(ordered_pair(U, V)) = U) & (second(ordered_pair(U, V)) = V)) | (~member(V, universal_class)) | (~member(U, universal_class))))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(5,axiom,(~![U: $i, V: $i, X: $i] : (((member(U, universal_class) & member(V, universal_class)) & (X = ordered_pair(U, V))) => ((first(X) = U) & (second(X) = V)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','unique_1st_and_2nd_in_pair_of_sets1')).
% 0.21/0.40 tff(6,plain,
% 0.21/0.40 (~![U: $i, V: $i] : (((first(ordered_pair(U, V)) = U) & (second(ordered_pair(U, V)) = V)) | (~member(V, universal_class)) | (~member(U, universal_class)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.21/0.40 tff(7,plain,
% 0.21/0.40 (~![U: $i, V: $i] : (((first(ordered_pair(U, V)) = U) & (second(ordered_pair(U, V)) = V)) | (~member(V, universal_class)) | (~member(U, universal_class)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[6, 3])).
% 0.21/0.40 tff(8,plain,
% 0.21/0.40 (~![U: $i, V: $i] : (((first(ordered_pair(U, V)) = U) & (second(ordered_pair(U, V)) = V)) | (~member(V, universal_class)) | (~member(U, universal_class)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.21/0.40 tff(9,plain,
% 0.21/0.40 (~![U: $i, V: $i] : (((first(ordered_pair(U, V)) = U) & (second(ordered_pair(U, V)) = V)) | (~member(V, universal_class)) | (~member(U, universal_class)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[8, 3])).
% 0.21/0.40 tff(10,plain,(
% 0.21/0.40 ~(((first(ordered_pair(U!8, V!7)) = U!8) & (second(ordered_pair(U!8, V!7)) = V!7)) | (~member(V!7, universal_class)) | (~member(U!8, universal_class)))),
% 0.21/0.40 inference(skolemize,[status(sab)],[9])).
% 0.21/0.40 tff(11,plain,
% 0.21/0.40 (~(((first(ordered_pair(U!8, V!7)) = U!8) & (second(ordered_pair(U!8, V!7)) = V!7)) | (~member(V!7, universal_class)) | (~member(U!8, universal_class)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[10, 2])).
% 0.21/0.40 tff(12,plain,
% 0.21/0.40 (member(U!8, universal_class)),
% 0.21/0.40 inference(or_elim,[status(thm)],[11])).
% 0.21/0.40 tff(13,plain,
% 0.21/0.40 (member(V!7, universal_class)),
% 0.21/0.40 inference(or_elim,[status(thm)],[11])).
% 0.21/0.40 tff(14,plain,
% 0.21/0.40 (^[X: $i, Y: $i] : refl(((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(15,plain,
% 0.21/0.40 (![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))) <=> ![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[14])).
% 0.21/0.41 tff(16,plain,
% 0.21/0.41 (^[X: $i, Y: $i] : trans(monotonicity(trans(monotonicity(rewrite((member(X, universal_class) & member(Y, universal_class)) <=> (~((~member(Y, universal_class)) | (~member(X, universal_class))))), ((~(member(X, universal_class) & member(Y, universal_class))) <=> (~(~((~member(Y, universal_class)) | (~member(X, universal_class))))))), rewrite((~(~((~member(Y, universal_class)) | (~member(X, universal_class))))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class)))), ((~(member(X, universal_class) & member(Y, universal_class))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class))))), rewrite(((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)) <=> (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))), (((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> (((~member(Y, universal_class)) | (~member(X, universal_class))) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))))), rewrite((((~member(Y, universal_class)) | (~member(X, universal_class))) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y))))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))), (((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(17,plain,
% 0.21/0.41 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[16])).
% 0.21/0.41 tff(18,plain,
% 0.21/0.41 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(19,plain,
% 0.21/0.41 (^[X: $i, Y: $i] : rewrite(((member(X, universal_class) & member(Y, universal_class)) => ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(20,plain,
% 0.21/0.41 (![X: $i, Y: $i] : ((member(X, universal_class) & member(Y, universal_class)) => ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y))) <=> ![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[19])).
% 0.21/0.41 tff(21,axiom,(![X: $i, Y: $i] : ((member(X, universal_class) & member(Y, universal_class)) => ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))), file('/export/starexec/sandbox/benchmark/Axioms/SET005+0.ax','first_second')).
% 0.21/0.41 tff(22,plain,
% 0.21/0.41 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[21, 20])).
% 0.21/0.41 tff(23,plain,
% 0.21/0.41 (![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[22, 18])).
% 0.21/0.41 tff(24,plain,(
% 0.21/0.41 ![X: $i, Y: $i] : ((~(member(X, universal_class) & member(Y, universal_class))) | ((first(ordered_pair(X, Y)) = X) & (second(ordered_pair(X, Y)) = Y)))),
% 0.21/0.41 inference(skolemize,[status(sab)],[23])).
% 0.21/0.41 tff(25,plain,
% 0.21/0.41 (![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[24, 17])).
% 0.21/0.41 tff(26,plain,
% 0.21/0.41 (![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[25, 15])).
% 0.21/0.41 tff(27,plain,
% 0.21/0.41 (((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | ((~member(V!7, universal_class)) | (~member(U!8, universal_class)) | (~((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7)))))) <=> ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | (~member(V!7, universal_class)) | (~member(U!8, universal_class)) | (~((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7)))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(28,plain,
% 0.21/0.41 ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | ((~member(V!7, universal_class)) | (~member(U!8, universal_class)) | (~((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7)))))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(29,plain,
% 0.21/0.41 ((~![X: $i, Y: $i] : ((~member(Y, universal_class)) | (~member(X, universal_class)) | (~((~(first(ordered_pair(X, Y)) = X)) | (~(second(ordered_pair(X, Y)) = Y)))))) | (~member(V!7, universal_class)) | (~member(U!8, universal_class)) | (~((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[28, 27])).
% 0.21/0.41 tff(30,plain,
% 0.21/0.41 ($false),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[29, 26, 13, 12, 1])).
% 0.21/0.41 tff(31,plain,(~((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7)))), inference(lemma,lemma(discharge,[]))).
% 0.21/0.41 tff(32,plain,
% 0.21/0.41 (((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7))) | (first(ordered_pair(U!8, V!7)) = U!8)),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(33,plain,
% 0.21/0.41 (first(ordered_pair(U!8, V!7)) = U!8),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[32, 31])).
% 0.21/0.41 tff(34,plain,
% 0.21/0.41 (((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7))) | (second(ordered_pair(U!8, V!7)) = V!7)),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(35,plain,
% 0.21/0.41 (second(ordered_pair(U!8, V!7)) = V!7),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[34, 31])).
% 0.21/0.41 tff(36,plain,
% 0.21/0.41 ((~(~((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7))))) <=> ((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(37,plain,
% 0.21/0.41 (((first(ordered_pair(U!8, V!7)) = U!8) & (second(ordered_pair(U!8, V!7)) = V!7)) <=> (~((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(38,plain,
% 0.21/0.41 ((~((first(ordered_pair(U!8, V!7)) = U!8) & (second(ordered_pair(U!8, V!7)) = V!7))) <=> (~(~((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7)))))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[37])).
% 0.21/0.41 tff(39,plain,
% 0.21/0.41 ((~((first(ordered_pair(U!8, V!7)) = U!8) & (second(ordered_pair(U!8, V!7)) = V!7))) <=> ((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7)))),
% 0.21/0.41 inference(transitivity,[status(thm)],[38, 36])).
% 0.21/0.41 tff(40,plain,
% 0.21/0.41 (~((first(ordered_pair(U!8, V!7)) = U!8) & (second(ordered_pair(U!8, V!7)) = V!7))),
% 0.21/0.41 inference(or_elim,[status(thm)],[11])).
% 0.21/0.41 tff(41,plain,
% 0.21/0.41 ((~(first(ordered_pair(U!8, V!7)) = U!8)) | (~(second(ordered_pair(U!8, V!7)) = V!7))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.21/0.41 tff(42,plain,
% 0.21/0.41 ($false),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[41, 35, 33])).
% 0.21/0.41 % SZS output end Proof
% 0.21/0.41 % E exiting
%------------------------------------------------------------------------------