%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET600+3 : TPTP v9.0.0. Released v2.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n015.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 05:25:57 AM UTC 2025
% Result : Theorem 0.12s 0.39s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET600+3 : TPTP v9.0.0. Released v2.2.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n015.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 05:30:40 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.12/0.39 % SZS status Theorem
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(member_type, type, (
% 0.12/0.39 member: ( $i * $i ) > $o)).
% 0.12/0.39 tff(union_type, type, (
% 0.12/0.39 union: ( $i * $i ) > $i)).
% 0.12/0.39 tff(empty_set_type, type, (
% 0.12/0.39 empty_set: $i)).
% 0.12/0.39 tff(tptp_fun_B_4_type, type, (
% 0.12/0.39 tptp_fun_B_4: $i)).
% 0.12/0.39 tff(tptp_fun_D_2_type, type, (
% 0.12/0.39 tptp_fun_D_2: ( $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_C_3_type, type, (
% 0.12/0.39 tptp_fun_C_3: $i)).
% 0.12/0.39 tff(1,plain,
% 0.12/0.39 (^[B: $i, C: $i] : refl((union(B, C) = union(C, B)) <=> (union(B, C) = union(C, B)))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(2,plain,
% 0.12/0.39 (![B: $i, C: $i] : (union(B, C) = union(C, B)) <=> ![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[1])).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 (![B: $i, C: $i] : (union(B, C) = union(C, B)) <=> ![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(4,axiom,(![B: $i, C: $i] : (union(B, C) = union(C, B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','commutativity_of_union')).
% 0.12/0.39 tff(5,plain,
% 0.12/0.39 (![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.12/0.39 tff(6,plain,(
% 0.12/0.39 ![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.12/0.39 inference(skolemize,[status(sab)],[5])).
% 0.12/0.39 tff(7,plain,
% 0.12/0.39 (![B: $i, C: $i] : (union(B, C) = union(C, B))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.12/0.39 tff(8,plain,
% 0.12/0.39 ((~![B: $i, C: $i] : (union(B, C) = union(C, B))) | (union(B!4, C!3) = union(C!3, B!4))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(9,plain,
% 0.12/0.39 (union(B!4, C!3) = union(C!3, B!4)),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[8, 7])).
% 0.12/0.39 tff(10,plain,
% 0.12/0.39 (^[B: $i, C: $i, D: $i] : refl((~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(11,plain,
% 0.12/0.39 (![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[10])).
% 0.12/0.39 tff(12,plain,
% 0.12/0.39 (![B: $i, C: $i] : ![D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.12/0.39 inference(pull_quant,[status(thm)],[])).
% 0.12/0.39 tff(13,plain,
% 0.12/0.39 (^[B: $i, C: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(pull_quant(((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) <=> ![D: $i] : ((~(B = C)) | (member(D, B) <=> member(D, C)))), ((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) <=> (~![D: $i] : ((~(B = C)) | (member(D, B) <=> member(D, C)))))), pull_quant((~![D: $i] : ((~(B = C)) | (member(D, B) <=> member(D, C)))) <=> ?[D: $i] : (~((~(B = C)) | (member(D, B) <=> member(D, C))))), ((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) <=> ?[D: $i] : (~((~(B = C)) | (member(D, B) <=> member(D, C)))))), (((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))) <=> (?[D: $i] : (~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))), pull_quant((?[D: $i] : (~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))) <=> ?[D: $i] : ((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))), (((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))) <=> ?[D: $i] : ((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))), ((~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> (~?[D: $i] : ((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))))), pull_quant((~?[D: $i] : ((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))), ((~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(14,plain,
% 0.12/0.39 (![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i] : ![D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[13])).
% 0.12/0.39 tff(15,plain,
% 0.12/0.39 (![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[14, 12])).
% 0.12/0.39 tff(16,plain,
% 0.12/0.39 (![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[15, 11])).
% 0.12/0.39 tff(17,plain,
% 0.12/0.39 (^[B: $i, C: $i] : rewrite((~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(18,plain,
% 0.12/0.39 (![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[17])).
% 0.12/0.39 tff(19,plain,
% 0.12/0.39 (![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))) <=> ![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[18, 16])).
% 0.12/0.39 tff(20,plain,
% 0.12/0.39 (^[B: $i, C: $i] : trans(monotonicity(rewrite(((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) <=> ((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))), trans(monotonicity(rewrite((~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))) <=> (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))), (((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))) <=> ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))), rewrite(((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))) <=> ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))), (((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))) <=> ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))), ((((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) & ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))) <=> (((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) & ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))))), rewrite((((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) & ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))) <=> (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))), ((((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) & ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))) <=> (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(21,plain,
% 0.12/0.39 (![B: $i, C: $i] : (((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) & ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C))))) <=> ![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[20])).
% 0.12/0.39 tff(22,plain,
% 0.12/0.39 (![B: $i, C: $i] : ((B = C) <=> ![D: $i] : (member(D, B) <=> member(D, C))) <=> ![B: $i, C: $i] : ((B = C) <=> ![D: $i] : (member(D, B) <=> member(D, C)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(23,axiom,(![B: $i, C: $i] : ((B = C) <=> ![D: $i] : (member(D, B) <=> member(D, C)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','equal_member_defn')).
% 0.12/0.39 tff(24,plain,
% 0.12/0.39 (![B: $i, C: $i] : ((B = C) <=> ![D: $i] : (member(D, B) <=> member(D, C)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.12/0.39 tff(25,plain,(
% 0.12/0.39 ![B: $i, C: $i] : (((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C))) & ((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))),
% 0.12/0.39 inference(skolemize,[status(sab)],[24])).
% 0.12/0.39 tff(26,plain,
% 0.12/0.39 (![B: $i, C: $i] : (~((~((~(B = C)) | ![D: $i] : (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[25, 21])).
% 0.12/0.39 tff(27,plain,
% 0.12/0.39 (![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[26, 19])).
% 0.12/0.39 tff(28,plain,
% 0.12/0.39 ((~![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))) | (~((~((~(union(C!3, B!4) = C!3)) | (member(union(C!3, B!4), union(C!3, B!4)) <=> member(union(C!3, B!4), C!3)))) | (~((union(C!3, B!4) = C!3) | (~(member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3)))))))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(29,plain,
% 0.12/0.39 (~((~((~(union(C!3, B!4) = C!3)) | (member(union(C!3, B!4), union(C!3, B!4)) <=> member(union(C!3, B!4), C!3)))) | (~((union(C!3, B!4) = C!3) | (~(member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3))))))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[28, 27])).
% 0.12/0.39 tff(30,plain,
% 0.12/0.39 (((~((~(union(C!3, B!4) = C!3)) | (member(union(C!3, B!4), union(C!3, B!4)) <=> member(union(C!3, B!4), C!3)))) | (~((union(C!3, B!4) = C!3) | (~(member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3)))))) | ((union(C!3, B!4) = C!3) | (~(member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3))))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(31,plain,
% 0.12/0.40 ((union(C!3, B!4) = C!3) | (~(member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3)))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[30, 29])).
% 0.12/0.40 tff(32,plain,
% 0.12/0.40 ((empty_set = C!3) <=> (C!3 = empty_set)),
% 0.12/0.40 inference(commutativity,[status(thm)],[])).
% 0.12/0.40 tff(33,assumption,(~(C!3 = empty_set)), introduced(assumption)).
% 0.12/0.40 tff(34,plain,
% 0.12/0.40 (((~(B!4 = empty_set)) | (~(C!3 = empty_set))) | (C!3 = empty_set)),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(35,plain,
% 0.12/0.40 ((~(B!4 = empty_set)) | (~(C!3 = empty_set))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[34, 33])).
% 0.12/0.40 tff(36,plain,
% 0.12/0.40 ((~(~(((~(B!4 = empty_set)) | (~(C!3 = empty_set))) <=> (union(B!4, C!3) = empty_set)))) <=> (((~(B!4 = empty_set)) | (~(C!3 = empty_set))) <=> (union(B!4, C!3) = empty_set))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(37,plain,
% 0.12/0.40 ((~(((~(B!4 = empty_set)) | (~(C!3 = empty_set))) <=> (union(B!4, C!3) = empty_set))) <=> (~(((~(B!4 = empty_set)) | (~(C!3 = empty_set))) <=> (union(B!4, C!3) = empty_set)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(38,plain,
% 0.12/0.40 (((union(B!4, C!3) = empty_set) <=> (~((~(B!4 = empty_set)) | (~(C!3 = empty_set))))) <=> (~(((~(B!4 = empty_set)) | (~(C!3 = empty_set))) <=> (union(B!4, C!3) = empty_set)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(39,plain,
% 0.12/0.40 (((B!4 = empty_set) & (C!3 = empty_set)) <=> (~((~(B!4 = empty_set)) | (~(C!3 = empty_set))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(40,plain,
% 0.12/0.40 (((union(B!4, C!3) = empty_set) <=> ((B!4 = empty_set) & (C!3 = empty_set))) <=> ((union(B!4, C!3) = empty_set) <=> (~((~(B!4 = empty_set)) | (~(C!3 = empty_set)))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[39])).
% 0.12/0.40 tff(41,plain,
% 0.12/0.40 (((union(B!4, C!3) = empty_set) <=> ((B!4 = empty_set) & (C!3 = empty_set))) <=> (~(((~(B!4 = empty_set)) | (~(C!3 = empty_set))) <=> (union(B!4, C!3) = empty_set)))),
% 0.12/0.40 inference(transitivity,[status(thm)],[40, 38])).
% 0.12/0.40 tff(42,plain,
% 0.12/0.40 (((union(B!4, C!3) = empty_set) <=> ((B!4 = empty_set) & (C!3 = empty_set))) <=> (~(((~(B!4 = empty_set)) | (~(C!3 = empty_set))) <=> (union(B!4, C!3) = empty_set)))),
% 0.12/0.40 inference(transitivity,[status(thm)],[41, 37])).
% 0.12/0.40 tff(43,plain,
% 0.12/0.40 ((~((union(B!4, C!3) = empty_set) <=> ((B!4 = empty_set) & (C!3 = empty_set)))) <=> (~(~(((~(B!4 = empty_set)) | (~(C!3 = empty_set))) <=> (union(B!4, C!3) = empty_set))))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[42])).
% 0.12/0.40 tff(44,plain,
% 0.12/0.40 ((~((union(B!4, C!3) = empty_set) <=> ((B!4 = empty_set) & (C!3 = empty_set)))) <=> (((~(B!4 = empty_set)) | (~(C!3 = empty_set))) <=> (union(B!4, C!3) = empty_set))),
% 0.12/0.40 inference(transitivity,[status(thm)],[43, 36])).
% 0.12/0.40 tff(45,plain,
% 0.12/0.40 ((~((union(B!4, C!3) = empty_set) <=> ((B!4 = empty_set) & (C!3 = empty_set)))) <=> (~((union(B!4, C!3) = empty_set) <=> ((B!4 = empty_set) & (C!3 = empty_set))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(46,plain,
% 0.12/0.40 ((~![B: $i, C: $i] : ((union(B, C) = empty_set) <=> ((B = empty_set) & (C = empty_set)))) <=> (~![B: $i, C: $i] : ((union(B, C) = empty_set) <=> ((B = empty_set) & (C = empty_set))))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(47,axiom,(~![B: $i, C: $i] : ((union(B, C) = empty_set) <=> ((B = empty_set) & (C = empty_set)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_th59')).
% 0.12/0.40 tff(48,plain,
% 0.12/0.40 (~![B: $i, C: $i] : ((union(B, C) = empty_set) <=> ((B = empty_set) & (C = empty_set)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.12/0.40 tff(49,plain,
% 0.12/0.40 (~![B: $i, C: $i] : ((union(B, C) = empty_set) <=> ((B = empty_set) & (C = empty_set)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[48, 46])).
% 0.12/0.40 tff(50,plain,
% 0.12/0.40 (~![B: $i, C: $i] : ((union(B, C) = empty_set) <=> ((B = empty_set) & (C = empty_set)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[49, 46])).
% 0.12/0.40 tff(51,plain,
% 0.12/0.40 (~![B: $i, C: $i] : ((union(B, C) = empty_set) <=> ((B = empty_set) & (C = empty_set)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[50, 46])).
% 0.12/0.40 tff(52,plain,(
% 0.12/0.40 ~((union(B!4, C!3) = empty_set) <=> ((B!4 = empty_set) & (C!3 = empty_set)))),
% 0.12/0.40 inference(skolemize,[status(sab)],[51])).
% 0.12/0.40 tff(53,plain,
% 0.12/0.40 (~((union(B!4, C!3) = empty_set) <=> ((B!4 = empty_set) & (C!3 = empty_set)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[52, 45])).
% 0.12/0.40 tff(54,plain,
% 0.12/0.40 (((~(B!4 = empty_set)) | (~(C!3 = empty_set))) <=> (union(B!4, C!3) = empty_set)),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[53, 44])).
% 0.12/0.40 tff(55,plain,
% 0.12/0.40 ((~((~(B!4 = empty_set)) | (~(C!3 = empty_set)))) | (union(B!4, C!3) = empty_set) | (~(((~(B!4 = empty_set)) | (~(C!3 = empty_set))) <=> (union(B!4, C!3) = empty_set)))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(56,plain,
% 0.12/0.40 ((~((~(B!4 = empty_set)) | (~(C!3 = empty_set)))) | (union(B!4, C!3) = empty_set)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[55, 54])).
% 0.12/0.40 tff(57,plain,
% 0.12/0.40 (union(B!4, C!3) = empty_set),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[56, 35])).
% 0.12/0.40 tff(58,plain,
% 0.12/0.40 (union(C!3, B!4) = union(B!4, C!3)),
% 0.12/0.40 inference(symmetry,[status(thm)],[9])).
% 0.12/0.40 tff(59,plain,
% 0.12/0.40 (union(C!3, B!4) = empty_set),
% 0.12/0.40 inference(transitivity,[status(thm)],[58, 57])).
% 0.12/0.40 tff(60,plain,
% 0.12/0.40 ((union(C!3, B!4) = C!3) <=> (empty_set = C!3)),
% 0.12/0.40 inference(monotonicity,[status(thm)],[59])).
% 0.12/0.40 tff(61,plain,
% 0.12/0.40 ((union(C!3, B!4) = C!3) <=> (C!3 = empty_set)),
% 0.12/0.40 inference(transitivity,[status(thm)],[60, 32])).
% 0.12/0.40 tff(62,plain,
% 0.12/0.40 ((C!3 = empty_set) <=> (union(C!3, B!4) = C!3)),
% 0.12/0.40 inference(symmetry,[status(thm)],[61])).
% 0.12/0.40 tff(63,plain,
% 0.12/0.40 ((~(C!3 = empty_set)) <=> (~(union(C!3, B!4) = C!3))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[62])).
% 0.12/0.40 tff(64,plain,
% 0.12/0.40 (~(union(C!3, B!4) = C!3)),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[33, 63])).
% 0.12/0.40 tff(65,plain,
% 0.12/0.40 ((~((union(C!3, B!4) = C!3) | (~(member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3))))) | (union(C!3, B!4) = C!3) | (~(member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3)))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(66,plain,
% 0.12/0.40 (~(member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[65, 64, 31])).
% 0.12/0.40 tff(67,plain,
% 0.12/0.40 (empty_set = union(B!4, C!3)),
% 0.12/0.40 inference(symmetry,[status(thm)],[57])).
% 0.12/0.40 tff(68,plain,
% 0.12/0.40 (empty_set = union(C!3, B!4)),
% 0.12/0.40 inference(transitivity,[status(thm)],[67, 9])).
% 0.12/0.40 tff(69,plain,
% 0.12/0.40 (member(tptp_fun_D_2(C!3, union(C!3, B!4)), empty_set) <=> member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[68])).
% 0.12/0.40 tff(70,plain,
% 0.12/0.40 ((~member(tptp_fun_D_2(C!3, union(C!3, B!4)), empty_set)) <=> (~member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)))),
% 0.12/0.40 inference(monotonicity,[status(thm)],[69])).
% 0.12/0.40 tff(71,plain,
% 0.12/0.40 (^[B: $i] : refl((~member(B, empty_set)) <=> (~member(B, empty_set)))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(72,plain,
% 0.12/0.40 (![B: $i] : (~member(B, empty_set)) <=> ![B: $i] : (~member(B, empty_set))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[71])).
% 0.12/0.40 tff(73,plain,
% 0.12/0.40 (![B: $i] : (~member(B, empty_set)) <=> ![B: $i] : (~member(B, empty_set))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(74,axiom,(![B: $i] : (~member(B, empty_set))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','empty_set_defn')).
% 0.12/0.40 tff(75,plain,
% 0.12/0.40 (![B: $i] : (~member(B, empty_set))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[74, 73])).
% 0.12/0.40 tff(76,plain,(
% 0.12/0.40 ![B: $i] : (~member(B, empty_set))),
% 0.12/0.40 inference(skolemize,[status(sab)],[75])).
% 0.12/0.40 tff(77,plain,
% 0.12/0.40 (![B: $i] : (~member(B, empty_set))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[76, 72])).
% 0.12/0.40 tff(78,plain,
% 0.12/0.40 ((~![B: $i] : (~member(B, empty_set))) | (~member(tptp_fun_D_2(C!3, union(C!3, B!4)), empty_set))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(79,plain,
% 0.12/0.40 (~member(tptp_fun_D_2(C!3, union(C!3, B!4)), empty_set)),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[78, 77])).
% 0.21/0.40 tff(80,plain,
% 0.21/0.40 (~member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[79, 70])).
% 0.21/0.40 tff(81,plain,
% 0.21/0.40 ((member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3)) | member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) | member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3)),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(82,plain,
% 0.21/0.40 (member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3)),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[81, 80, 66])).
% 0.21/0.40 tff(83,plain,
% 0.21/0.40 (^[B: $i, C: $i, D: $i] : refl((member(D, union(B, C)) <=> (member(D, C) | member(D, B))) <=> (member(D, union(B, C)) <=> (member(D, C) | member(D, B))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(84,plain,
% 0.21/0.40 (![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B))) <=> ![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[83])).
% 0.21/0.40 tff(85,plain,
% 0.21/0.40 (![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B))) <=> ![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(86,plain,
% 0.21/0.40 (^[B: $i, C: $i, D: $i] : rewrite((member(D, union(B, C)) <=> (member(D, B) | member(D, C))) <=> (member(D, union(B, C)) <=> (member(D, C) | member(D, B))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(87,plain,
% 0.21/0.40 (![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, B) | member(D, C))) <=> ![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[86])).
% 0.21/0.40 tff(88,axiom,(![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, B) | member(D, C)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','union_defn')).
% 0.21/0.40 tff(89,plain,
% 0.21/0.40 (![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.21/0.40 tff(90,plain,
% 0.21/0.40 (![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[89, 85])).
% 0.21/0.40 tff(91,plain,(
% 0.21/0.40 ![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.21/0.40 inference(skolemize,[status(sab)],[90])).
% 0.21/0.40 tff(92,plain,
% 0.21/0.40 (![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[91, 84])).
% 0.21/0.40 tff(93,plain,
% 0.21/0.40 ((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) <=> (member(tptp_fun_D_2(C!3, union(C!3, B!4)), B!4) | member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3)))),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(94,plain,
% 0.21/0.40 (member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) <=> (member(tptp_fun_D_2(C!3, union(C!3, B!4)), B!4) | member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[93, 92])).
% 0.21/0.40 tff(95,plain,
% 0.21/0.40 ((~(member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) <=> (member(tptp_fun_D_2(C!3, union(C!3, B!4)), B!4) | member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3)))) | member(tptp_fun_D_2(C!3, union(C!3, B!4)), union(C!3, B!4)) | (~(member(tptp_fun_D_2(C!3, union(C!3, B!4)), B!4) | member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3)))),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(96,plain,
% 0.21/0.40 (~(member(tptp_fun_D_2(C!3, union(C!3, B!4)), B!4) | member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[95, 80, 94])).
% 0.21/0.40 tff(97,plain,
% 0.21/0.40 ((member(tptp_fun_D_2(C!3, union(C!3, B!4)), B!4) | member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3)) | (~member(tptp_fun_D_2(C!3, union(C!3, B!4)), C!3))),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(98,plain,
% 0.21/0.40 ($false),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[97, 96, 82])).
% 0.21/0.40 tff(99,plain,(C!3 = empty_set), inference(lemma,lemma(discharge,[]))).
% 0.21/0.40 tff(100,plain,
% 0.21/0.40 (empty_set = C!3),
% 0.21/0.40 inference(symmetry,[status(thm)],[99])).
% 0.21/0.40 tff(101,assumption,(~(union(B!4, C!3) = empty_set)), introduced(assumption)).
% 0.21/0.40 tff(102,plain,
% 0.21/0.40 (~((~(B!4 = empty_set)) | (~(C!3 = empty_set)))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[56, 101])).
% 0.21/0.40 tff(103,plain,
% 0.21/0.40 (((~(B!4 = empty_set)) | (~(C!3 = empty_set))) | (B!4 = empty_set)),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(104,plain,
% 0.21/0.40 (B!4 = empty_set),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[103, 102])).
% 0.21/0.40 tff(105,plain,
% 0.21/0.40 (empty_set = B!4),
% 0.21/0.40 inference(symmetry,[status(thm)],[104])).
% 0.21/0.40 tff(106,plain,
% 0.21/0.40 (union(empty_set, empty_set) = union(B!4, C!3)),
% 0.21/0.40 inference(monotonicity,[status(thm)],[105, 100])).
% 0.21/0.40 tff(107,plain,
% 0.21/0.40 (union(empty_set, empty_set) = union(C!3, B!4)),
% 0.21/0.40 inference(transitivity,[status(thm)],[106, 9])).
% 0.21/0.40 tff(108,plain,
% 0.21/0.40 (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4))),
% 0.21/0.40 inference(monotonicity,[status(thm)],[107])).
% 0.21/0.41 tff(109,plain,
% 0.21/0.41 (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set))),
% 0.21/0.41 inference(symmetry,[status(thm)],[108])).
% 0.21/0.41 tff(110,plain,
% 0.21/0.41 ((union(C!3, B!4) = empty_set) <=> (union(B!4, C!3) = empty_set)),
% 0.21/0.41 inference(monotonicity,[status(thm)],[58])).
% 0.21/0.41 tff(111,plain,
% 0.21/0.41 ((union(B!4, C!3) = empty_set) <=> (union(C!3, B!4) = empty_set)),
% 0.21/0.41 inference(symmetry,[status(thm)],[110])).
% 0.21/0.41 tff(112,plain,
% 0.21/0.41 ((~(union(B!4, C!3) = empty_set)) <=> (~(union(C!3, B!4) = empty_set))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[111])).
% 0.21/0.41 tff(113,plain,
% 0.21/0.41 (~(union(C!3, B!4) = empty_set)),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[101, 112])).
% 0.21/0.41 tff(114,plain,
% 0.21/0.41 ((~![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))) | (~((~((~(union(C!3, B!4) = empty_set)) | (member(empty_set, union(C!3, B!4)) <=> member(empty_set, empty_set)))) | (~((union(C!3, B!4) = empty_set) | (~(member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)))))))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(115,plain,
% 0.21/0.41 (~((~((~(union(C!3, B!4) = empty_set)) | (member(empty_set, union(C!3, B!4)) <=> member(empty_set, empty_set)))) | (~((union(C!3, B!4) = empty_set) | (~(member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set))))))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[114, 27])).
% 0.21/0.41 tff(116,plain,
% 0.21/0.41 (((~((~(union(C!3, B!4) = empty_set)) | (member(empty_set, union(C!3, B!4)) <=> member(empty_set, empty_set)))) | (~((union(C!3, B!4) = empty_set) | (~(member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)))))) | ((union(C!3, B!4) = empty_set) | (~(member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set))))),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(117,plain,
% 0.21/0.41 ((union(C!3, B!4) = empty_set) | (~(member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[116, 115])).
% 0.21/0.41 tff(118,plain,
% 0.21/0.41 ((~((union(C!3, B!4) = empty_set) | (~(member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set))))) | (union(C!3, B!4) = empty_set) | (~(member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)))),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(119,plain,
% 0.21/0.41 ((union(C!3, B!4) = empty_set) | (~(member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[118, 117])).
% 0.21/0.41 tff(120,plain,
% 0.21/0.41 (~(member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[119, 113])).
% 0.21/0.41 tff(121,plain,
% 0.21/0.41 ((~![B: $i] : (~member(B, empty_set))) | (~member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(122,plain,
% 0.21/0.41 (~member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[121, 77])).
% 0.21/0.41 tff(123,plain,
% 0.21/0.41 ((member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)) | member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4)) | member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(124,plain,
% 0.21/0.41 (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(C!3, B!4))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[123, 122, 120])).
% 0.21/0.41 tff(125,plain,
% 0.21/0.41 (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[124, 109])).
% 0.21/0.41 tff(126,plain,
% 0.21/0.41 (((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set))) <=> ((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(127,plain,
% 0.21/0.41 ((member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set) | member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set))) <=> (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(128,plain,
% 0.21/0.41 (((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set) | member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)))) <=> ((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[127])).
% 0.21/0.41 tff(129,plain,
% 0.21/0.41 (((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set) | member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)))) <=> ((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)))),
% 0.21/0.41 inference(transitivity,[status(thm)],[128, 126])).
% 0.21/0.41 tff(130,plain,
% 0.21/0.41 ((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set) | member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(131,plain,
% 0.21/0.41 ((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[130, 129])).
% 0.21/0.41 tff(132,plain,
% 0.21/0.41 (member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[131, 92])).
% 0.21/0.41 tff(133,plain,
% 0.21/0.41 ((~(member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set))) | (~member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set))) | member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set)),
% 0.21/0.41 inference(tautology,[status(thm)],[])).
% 0.21/0.41 tff(134,plain,
% 0.21/0.41 ((~(member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)) <=> member(tptp_fun_D_2(empty_set, union(C!3, B!4)), empty_set))) | (~member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set)))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[133, 122])).
% 0.21/0.41 tff(135,plain,
% 0.21/0.41 (~member(tptp_fun_D_2(empty_set, union(C!3, B!4)), union(empty_set, empty_set))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[134, 132])).
% 0.21/0.41 tff(136,plain,
% 0.21/0.41 ($false),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[135, 125])).
% 0.21/0.41 tff(137,plain,(union(B!4, C!3) = empty_set), inference(lemma,lemma(discharge,[]))).
% 0.21/0.41 tff(138,plain,
% 0.21/0.41 (union(B!4, empty_set) = union(B!4, C!3)),
% 0.21/0.41 inference(monotonicity,[status(thm)],[100])).
% 0.21/0.41 tff(139,plain,
% 0.21/0.41 (union(B!4, empty_set) = empty_set),
% 0.21/0.41 inference(transitivity,[status(thm)],[138, 137])).
% 0.21/0.41 tff(140,plain,
% 0.21/0.41 (member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) <=> member(tptp_fun_D_2(empty_set, B!4), empty_set)),
% 0.21/0.41 inference(monotonicity,[status(thm)],[139])).
% 0.21/0.41 tff(141,plain,
% 0.21/0.41 (member(tptp_fun_D_2(empty_set, B!4), empty_set) <=> member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set))),
% 0.21/0.41 inference(symmetry,[status(thm)],[140])).
% 0.21/0.41 tff(142,plain,
% 0.21/0.41 ((~member(tptp_fun_D_2(empty_set, B!4), empty_set)) <=> (~member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[141])).
% 0.21/0.41 tff(143,plain,
% 0.21/0.41 ((~![B: $i] : (~member(B, empty_set))) | (~member(tptp_fun_D_2(empty_set, B!4), empty_set))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(144,plain,
% 0.21/0.41 (~member(tptp_fun_D_2(empty_set, B!4), empty_set)),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[143, 77])).
% 0.21/0.41 tff(145,plain,
% 0.21/0.41 (~member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[144, 142])).
% 0.21/0.41 tff(146,plain,
% 0.21/0.41 (((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) <=> (member(tptp_fun_D_2(empty_set, B!4), B!4) | member(tptp_fun_D_2(empty_set, B!4), empty_set)))) <=> ((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) <=> (member(tptp_fun_D_2(empty_set, B!4), B!4) | member(tptp_fun_D_2(empty_set, B!4), empty_set))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(147,plain,
% 0.21/0.41 ((member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) <=> (member(tptp_fun_D_2(empty_set, B!4), empty_set) | member(tptp_fun_D_2(empty_set, B!4), B!4))) <=> (member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) <=> (member(tptp_fun_D_2(empty_set, B!4), B!4) | member(tptp_fun_D_2(empty_set, B!4), empty_set)))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(148,plain,
% 0.21/0.41 (((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) <=> (member(tptp_fun_D_2(empty_set, B!4), empty_set) | member(tptp_fun_D_2(empty_set, B!4), B!4)))) <=> ((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) <=> (member(tptp_fun_D_2(empty_set, B!4), B!4) | member(tptp_fun_D_2(empty_set, B!4), empty_set))))),
% 0.21/0.41 inference(monotonicity,[status(thm)],[147])).
% 0.21/0.41 tff(149,plain,
% 0.21/0.41 (((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) <=> (member(tptp_fun_D_2(empty_set, B!4), empty_set) | member(tptp_fun_D_2(empty_set, B!4), B!4)))) <=> ((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) <=> (member(tptp_fun_D_2(empty_set, B!4), B!4) | member(tptp_fun_D_2(empty_set, B!4), empty_set))))),
% 0.21/0.42 inference(transitivity,[status(thm)],[148, 146])).
% 0.21/0.42 tff(150,plain,
% 0.21/0.42 ((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) <=> (member(tptp_fun_D_2(empty_set, B!4), empty_set) | member(tptp_fun_D_2(empty_set, B!4), B!4)))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(151,plain,
% 0.21/0.42 ((~![B: $i, C: $i, D: $i] : (member(D, union(B, C)) <=> (member(D, C) | member(D, B)))) | (member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) <=> (member(tptp_fun_D_2(empty_set, B!4), B!4) | member(tptp_fun_D_2(empty_set, B!4), empty_set)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[150, 149])).
% 0.21/0.42 tff(152,plain,
% 0.21/0.42 (member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) <=> (member(tptp_fun_D_2(empty_set, B!4), B!4) | member(tptp_fun_D_2(empty_set, B!4), empty_set))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[151, 92])).
% 0.21/0.42 tff(153,plain,
% 0.21/0.42 ((~![B: $i, C: $i, D: $i] : (~((~((~(B = C)) | (member(D, B) <=> member(D, C)))) | (~((B = C) | (~(member(tptp_fun_D_2(C, B), B) <=> member(tptp_fun_D_2(C, B), C)))))))) | (~((~((~(B!4 = empty_set)) | (member(empty_set, B!4) <=> member(empty_set, empty_set)))) | (~((B!4 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!4), B!4) <=> member(tptp_fun_D_2(empty_set, B!4), empty_set)))))))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(154,plain,
% 0.21/0.42 (~((~((~(B!4 = empty_set)) | (member(empty_set, B!4) <=> member(empty_set, empty_set)))) | (~((B!4 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!4), B!4) <=> member(tptp_fun_D_2(empty_set, B!4), empty_set))))))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[153, 27])).
% 0.21/0.42 tff(155,plain,
% 0.21/0.42 (((~((~(B!4 = empty_set)) | (member(empty_set, B!4) <=> member(empty_set, empty_set)))) | (~((B!4 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!4), B!4) <=> member(tptp_fun_D_2(empty_set, B!4), empty_set)))))) | ((B!4 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!4), B!4) <=> member(tptp_fun_D_2(empty_set, B!4), empty_set))))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(156,plain,
% 0.21/0.42 ((B!4 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!4), B!4) <=> member(tptp_fun_D_2(empty_set, B!4), empty_set)))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[155, 154])).
% 0.21/0.42 tff(157,plain,
% 0.21/0.42 (((~(B!4 = empty_set)) | (~(C!3 = empty_set))) | (~(union(B!4, C!3) = empty_set)) | (~(((~(B!4 = empty_set)) | (~(C!3 = empty_set))) <=> (union(B!4, C!3) = empty_set)))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(158,plain,
% 0.21/0.42 (((~(B!4 = empty_set)) | (~(C!3 = empty_set))) | (~(union(B!4, C!3) = empty_set))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[157, 54])).
% 0.21/0.42 tff(159,plain,
% 0.21/0.42 ((~(B!4 = empty_set)) | (~(C!3 = empty_set))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[158, 137])).
% 0.21/0.42 tff(160,plain,
% 0.21/0.42 ((~((~(B!4 = empty_set)) | (~(C!3 = empty_set)))) | (~(B!4 = empty_set)) | (~(C!3 = empty_set))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(161,plain,
% 0.21/0.42 ((~((~(B!4 = empty_set)) | (~(C!3 = empty_set)))) | (~(B!4 = empty_set))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[160, 99])).
% 0.21/0.42 tff(162,plain,
% 0.21/0.42 (~(B!4 = empty_set)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[161, 159])).
% 0.21/0.42 tff(163,plain,
% 0.21/0.42 ((~((B!4 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!4), B!4) <=> member(tptp_fun_D_2(empty_set, B!4), empty_set))))) | (B!4 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!4), B!4) <=> member(tptp_fun_D_2(empty_set, B!4), empty_set)))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(164,plain,
% 0.21/0.42 ((~((B!4 = empty_set) | (~(member(tptp_fun_D_2(empty_set, B!4), B!4) <=> member(tptp_fun_D_2(empty_set, B!4), empty_set))))) | (~(member(tptp_fun_D_2(empty_set, B!4), B!4) <=> member(tptp_fun_D_2(empty_set, B!4), empty_set)))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[163, 162])).
% 0.21/0.42 tff(165,plain,
% 0.21/0.42 (~(member(tptp_fun_D_2(empty_set, B!4), B!4) <=> member(tptp_fun_D_2(empty_set, B!4), empty_set))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[164, 156])).
% 0.21/0.42 tff(166,plain,
% 0.21/0.42 ((member(tptp_fun_D_2(empty_set, B!4), B!4) <=> member(tptp_fun_D_2(empty_set, B!4), empty_set)) | member(tptp_fun_D_2(empty_set, B!4), B!4) | member(tptp_fun_D_2(empty_set, B!4), empty_set)),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(167,plain,
% 0.21/0.42 (member(tptp_fun_D_2(empty_set, B!4), B!4)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[166, 144, 165])).
% 0.21/0.42 tff(168,plain,
% 0.21/0.42 ((member(tptp_fun_D_2(empty_set, B!4), B!4) | member(tptp_fun_D_2(empty_set, B!4), empty_set)) | (~member(tptp_fun_D_2(empty_set, B!4), B!4))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(169,plain,
% 0.21/0.42 (member(tptp_fun_D_2(empty_set, B!4), B!4) | member(tptp_fun_D_2(empty_set, B!4), empty_set)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[168, 167])).
% 0.21/0.42 tff(170,plain,
% 0.21/0.42 ((~(member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) <=> (member(tptp_fun_D_2(empty_set, B!4), B!4) | member(tptp_fun_D_2(empty_set, B!4), empty_set)))) | member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set)) | (~(member(tptp_fun_D_2(empty_set, B!4), B!4) | member(tptp_fun_D_2(empty_set, B!4), empty_set)))),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(171,plain,
% 0.21/0.42 (member(tptp_fun_D_2(empty_set, B!4), union(B!4, empty_set))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[170, 169, 152])).
% 0.21/0.42 tff(172,plain,
% 0.21/0.42 ($false),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[171, 145])).
% 0.21/0.42 % SZS output end Proof
% 0.21/0.42 % E exiting
%------------------------------------------------------------------------------