%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET930+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n012.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 05:26:41 AM UTC 2025
% Result : Theorem 0.24s 0.41s
% Output : Proof 0.24s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.15 % Problem : SET930+1 : TPTP v9.0.0. Released v3.2.0.
% 0.04/0.15 % Command : run_E %s %d THM
% 0.15/0.37 % Computer : n012.cluster.edu
% 0.15/0.37 % Model : x86_64 x86_64
% 0.15/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.37 % Memory : 8042.1875MB
% 0.15/0.37 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.37 % CPULimit : 300
% 0.15/0.37 % WCLimit : 300
% 0.15/0.37 % DateTime : Fri Jun 20 05:50:10 EDT 2025
% 0.15/0.37 % CPUTime :
% 0.24/0.41 % SZS status Theorem
% 0.24/0.41 % SZS output start Proof
% 0.24/0.41 tff(in_type, type, (
% 0.24/0.41 in: ( $i * $i ) > $o)).
% 0.24/0.41 tff(tptp_fun_C_2_type, type, (
% 0.24/0.41 tptp_fun_C_2: $i)).
% 0.24/0.41 tff(tptp_fun_B_3_type, type, (
% 0.24/0.41 tptp_fun_B_3: $i)).
% 0.24/0.41 tff(tptp_fun_A_4_type, type, (
% 0.24/0.41 tptp_fun_A_4: $i)).
% 0.24/0.41 tff(singleton_type, type, (
% 0.24/0.41 singleton: $i > $i)).
% 0.24/0.41 tff(set_difference_type, type, (
% 0.24/0.41 set_difference: ( $i * $i ) > $i)).
% 0.24/0.41 tff(unordered_pair_type, type, (
% 0.24/0.41 unordered_pair: ( $i * $i ) > $i)).
% 0.24/0.41 tff(empty_set_type, type, (
% 0.24/0.41 empty_set: $i)).
% 0.24/0.41 tff(1,assumption,(~in(A!4, C!2)), introduced(assumption)).
% 0.24/0.41 tff(2,plain,
% 0.24/0.41 (^[A: $i, B: $i, C: $i] : refl((~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A)))) <=> (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A)))))),
% 0.24/0.41 inference(bind,[status(th)],[])).
% 0.24/0.41 tff(3,plain,
% 0.24/0.41 (![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A)))) <=> ![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))),
% 0.24/0.41 inference(quant_intro,[status(thm)],[2])).
% 0.24/0.41 tff(4,plain,
% 0.24/0.41 (^[A: $i, B: $i, C: $i] : trans(trans(monotonicity(trans(monotonicity(rewrite((in(B, C) | (A = B)) <=> (in(B, C) | (A = B))), (((~in(A, C)) & (in(B, C) | (A = B))) <=> ((~in(A, C)) & (in(B, C) | (A = B))))), rewrite(((~in(A, C)) & (in(B, C) | (A = B))) <=> (~(in(A, C) | (~(in(B, C) | (A = B)))))), (((~in(A, C)) & (in(B, C) | (A = B))) <=> (~(in(A, C) | (~(in(B, C) | (A = B))))))), (((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B)))) <=> ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> (~(in(A, C) | (~(in(B, C) | (A = B)))))))), rewrite(((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> (~(in(A, C) | (~(in(B, C) | (A = B)))))) <=> (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))), (((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B)))) <=> (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A)))))), rewrite((~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A)))) <=> (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))), (((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B)))) <=> (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))))),
% 0.24/0.41 inference(bind,[status(th)],[])).
% 0.24/0.41 tff(5,plain,
% 0.24/0.41 (![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B)))) <=> ![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))),
% 0.24/0.41 inference(quant_intro,[status(thm)],[4])).
% 0.24/0.41 tff(6,plain,
% 0.24/0.41 (![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B)))) <=> ![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B))))),
% 0.24/0.41 inference(rewrite,[status(thm)],[])).
% 0.24/0.41 tff(7,axiom,(![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','l39_zfmisc_1')).
% 0.24/0.41 tff(8,plain,
% 0.24/0.41 (![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B))))),
% 0.24/0.41 inference(modus_ponens,[status(thm)],[7, 6])).
% 0.24/0.41 tff(9,plain,(
% 0.24/0.41 ![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = singleton(A)) <=> ((~in(A, C)) & (in(B, C) | (A = B))))),
% 0.24/0.41 inference(skolemize,[status(sab)],[8])).
% 0.24/0.41 tff(10,plain,
% 0.24/0.41 (![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))),
% 0.24/0.41 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.24/0.41 tff(11,plain,
% 0.24/0.41 (![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))),
% 0.24/0.41 inference(modus_ponens,[status(thm)],[10, 3])).
% 0.24/0.41 tff(12,plain,
% 0.24/0.41 ((~![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))) | (~((in(A!4, C!2) | (~(in(B!3, C!2) | (A!4 = B!3)))) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = singleton(A!4))))),
% 0.24/0.41 inference(quant_inst,[status(thm)],[])).
% 0.24/0.41 tff(13,plain,
% 0.24/0.41 (~((in(A!4, C!2) | (~(in(B!3, C!2) | (A!4 = B!3)))) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = singleton(A!4)))),
% 0.24/0.41 inference(unit_resolution,[status(thm)],[12, 11])).
% 0.24/0.41 tff(14,plain,
% 0.24/0.41 ((~![A: $i, B: $i, C: $i] : (~((~(set_difference(unordered_pair(A, B), C) = empty_set)) & (~(set_difference(unordered_pair(A, B), C) = singleton(A))) & (~(set_difference(unordered_pair(A, B), C) = singleton(B))) & (~(set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)))))) <=> (~![A: $i, B: $i, C: $i] : (~((~(set_difference(unordered_pair(A, B), C) = empty_set)) & (~(set_difference(unordered_pair(A, B), C) = singleton(A))) & (~(set_difference(unordered_pair(A, B), C) = singleton(B))) & (~(set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))))),
% 0.24/0.41 inference(rewrite,[status(thm)],[])).
% 0.24/0.41 tff(15,plain,
% 0.24/0.41 ((~![A: $i, B: $i, C: $i] : (~((((~(set_difference(unordered_pair(A, B), C) = empty_set)) & (~(set_difference(unordered_pair(A, B), C) = singleton(A)))) & (~(set_difference(unordered_pair(A, B), C) = singleton(B)))) & (~(set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)))))) <=> (~![A: $i, B: $i, C: $i] : (~((~(set_difference(unordered_pair(A, B), C) = empty_set)) & (~(set_difference(unordered_pair(A, B), C) = singleton(A))) & (~(set_difference(unordered_pair(A, B), C) = singleton(B))) & (~(set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))))),
% 0.24/0.41 inference(rewrite,[status(thm)],[])).
% 0.24/0.41 tff(16,axiom,(~![A: $i, B: $i, C: $i] : (~((((~(set_difference(unordered_pair(A, B), C) = empty_set)) & (~(set_difference(unordered_pair(A, B), C) = singleton(A)))) & (~(set_difference(unordered_pair(A, B), C) = singleton(B)))) & (~(set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t74_zfmisc_1')).
% 0.24/0.41 tff(17,plain,
% 0.24/0.41 (~![A: $i, B: $i, C: $i] : (~((~(set_difference(unordered_pair(A, B), C) = empty_set)) & (~(set_difference(unordered_pair(A, B), C) = singleton(A))) & (~(set_difference(unordered_pair(A, B), C) = singleton(B))) & (~(set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)))))),
% 0.24/0.41 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.24/0.41 tff(18,plain,
% 0.24/0.41 (~![A: $i, B: $i, C: $i] : (~((~(set_difference(unordered_pair(A, B), C) = empty_set)) & (~(set_difference(unordered_pair(A, B), C) = singleton(A))) & (~(set_difference(unordered_pair(A, B), C) = singleton(B))) & (~(set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)))))),
% 0.24/0.41 inference(modus_ponens,[status(thm)],[17, 14])).
% 0.24/0.41 tff(19,plain,
% 0.24/0.41 (~![A: $i, B: $i, C: $i] : (~((~(set_difference(unordered_pair(A, B), C) = empty_set)) & (~(set_difference(unordered_pair(A, B), C) = singleton(A))) & (~(set_difference(unordered_pair(A, B), C) = singleton(B))) & (~(set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)))))),
% 0.24/0.41 inference(modus_ponens,[status(thm)],[18, 14])).
% 0.24/0.41 tff(20,plain,
% 0.24/0.41 (~![A: $i, B: $i, C: $i] : (~((~(set_difference(unordered_pair(A, B), C) = empty_set)) & (~(set_difference(unordered_pair(A, B), C) = singleton(A))) & (~(set_difference(unordered_pair(A, B), C) = singleton(B))) & (~(set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)))))),
% 0.24/0.41 inference(modus_ponens,[status(thm)],[19, 14])).
% 0.24/0.41 tff(21,plain,(
% 0.24/0.41 (~(set_difference(unordered_pair(A!4, B!3), C!2) = empty_set)) & (~(set_difference(unordered_pair(A!4, B!3), C!2) = singleton(A!4))) & (~(set_difference(unordered_pair(A!4, B!3), C!2) = singleton(B!3))) & (~(set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3)))),
% 0.24/0.41 inference(skolemize,[status(sab)],[20])).
% 0.24/0.41 tff(22,plain,
% 0.24/0.41 (~(set_difference(unordered_pair(A!4, B!3), C!2) = singleton(A!4))),
% 0.24/0.41 inference(and_elim,[status(thm)],[21])).
% 0.24/0.41 tff(23,plain,
% 0.24/0.41 (((in(A!4, C!2) | (~(in(B!3, C!2) | (A!4 = B!3)))) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = singleton(A!4))) | (in(A!4, C!2) | (~(in(B!3, C!2) | (A!4 = B!3)))) | (set_difference(unordered_pair(A!4, B!3), C!2) = singleton(A!4))),
% 0.24/0.41 inference(tautology,[status(thm)],[])).
% 0.24/0.41 tff(24,plain,
% 0.24/0.41 (((in(A!4, C!2) | (~(in(B!3, C!2) | (A!4 = B!3)))) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = singleton(A!4))) | (in(A!4, C!2) | (~(in(B!3, C!2) | (A!4 = B!3))))),
% 0.24/0.41 inference(unit_resolution,[status(thm)],[23, 22])).
% 0.24/0.41 tff(25,plain,
% 0.24/0.41 (in(A!4, C!2) | (~(in(B!3, C!2) | (A!4 = B!3)))),
% 0.24/0.41 inference(unit_resolution,[status(thm)],[24, 13])).
% 0.24/0.41 tff(26,plain,
% 0.24/0.41 ((~(in(A!4, C!2) | (~(in(B!3, C!2) | (A!4 = B!3))))) | in(A!4, C!2) | (~(in(B!3, C!2) | (A!4 = B!3)))),
% 0.24/0.41 inference(tautology,[status(thm)],[])).
% 0.24/0.41 tff(27,plain,
% 0.24/0.41 (in(A!4, C!2) | (~(in(B!3, C!2) | (A!4 = B!3)))),
% 0.24/0.41 inference(unit_resolution,[status(thm)],[26, 25])).
% 0.24/0.41 tff(28,plain,
% 0.24/0.41 (~(in(B!3, C!2) | (A!4 = B!3))),
% 0.24/0.41 inference(unit_resolution,[status(thm)],[27, 1])).
% 0.24/0.41 tff(29,plain,
% 0.24/0.41 (^[A: $i, B: $i, C: $i] : refl((~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)))) <=> (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)))))),
% 0.24/0.41 inference(bind,[status(th)],[])).
% 0.24/0.41 tff(30,plain,
% 0.24/0.41 (![A: $i, B: $i, C: $i] : (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)))) <=> ![A: $i, B: $i, C: $i] : (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))),
% 0.24/0.41 inference(quant_intro,[status(thm)],[29])).
% 0.24/0.41 tff(31,plain,
% 0.24/0.41 (^[A: $i, B: $i, C: $i] : trans(trans(monotonicity(rewrite(((~in(A, C)) & (~in(B, C))) <=> (~(in(B, C) | in(A, C)))), (((set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)) <=> ((~in(A, C)) & (~in(B, C)))) <=> ((set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)) <=> (~(in(B, C) | in(A, C)))))), rewrite(((set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)) <=> (~(in(B, C) | in(A, C)))) <=> (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))), (((set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)) <=> ((~in(A, C)) & (~in(B, C)))) <=> (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)))))), rewrite((~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)))) <=> (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))), (((set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)) <=> ((~in(A, C)) & (~in(B, C)))) <=> (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))))),
% 0.24/0.41 inference(bind,[status(th)],[])).
% 0.24/0.41 tff(32,plain,
% 0.24/0.41 (![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)) <=> ((~in(A, C)) & (~in(B, C)))) <=> ![A: $i, B: $i, C: $i] : (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))),
% 0.24/0.41 inference(quant_intro,[status(thm)],[31])).
% 0.24/0.41 tff(33,plain,
% 0.24/0.41 (![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)) <=> ((~in(A, C)) & (~in(B, C)))) <=> ![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)) <=> ((~in(A, C)) & (~in(B, C))))),
% 0.24/0.41 inference(rewrite,[status(thm)],[])).
% 0.24/0.41 tff(34,axiom,(![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)) <=> ((~in(A, C)) & (~in(B, C))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t72_zfmisc_1')).
% 0.24/0.41 tff(35,plain,
% 0.24/0.41 (![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)) <=> ((~in(A, C)) & (~in(B, C))))),
% 0.24/0.41 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.24/0.42 tff(36,plain,(
% 0.24/0.42 ![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = unordered_pair(A, B)) <=> ((~in(A, C)) & (~in(B, C))))),
% 0.24/0.42 inference(skolemize,[status(sab)],[35])).
% 0.24/0.42 tff(37,plain,
% 0.24/0.42 (![A: $i, B: $i, C: $i] : (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))),
% 0.24/0.42 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.24/0.42 tff(38,plain,
% 0.24/0.42 (![A: $i, B: $i, C: $i] : (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))),
% 0.24/0.42 inference(modus_ponens,[status(thm)],[37, 30])).
% 0.24/0.42 tff(39,plain,
% 0.24/0.42 (((~![A: $i, B: $i, C: $i] : (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))) | (~((in(A!4, C!2) | in(B!3, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3))))) <=> ((~![A: $i, B: $i, C: $i] : (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))) | (~((in(A!4, C!2) | in(B!3, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3)))))),
% 0.24/0.42 inference(rewrite,[status(thm)],[])).
% 0.24/0.42 tff(40,plain,
% 0.24/0.42 ((~((in(A!4, C!2) | in(B!3, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3)))) <=> (~((in(A!4, C!2) | in(B!3, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3))))),
% 0.24/0.42 inference(rewrite,[status(thm)],[])).
% 0.24/0.42 tff(41,plain,
% 0.24/0.42 (((in(B!3, C!2) | in(A!4, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3))) <=> ((in(A!4, C!2) | in(B!3, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3)))),
% 0.24/0.42 inference(rewrite,[status(thm)],[])).
% 0.24/0.42 tff(42,plain,
% 0.24/0.42 ((~((in(B!3, C!2) | in(A!4, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3)))) <=> (~((in(A!4, C!2) | in(B!3, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3))))),
% 0.24/0.42 inference(monotonicity,[status(thm)],[41])).
% 0.24/0.42 tff(43,plain,
% 0.24/0.42 ((~((in(B!3, C!2) | in(A!4, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3)))) <=> (~((in(A!4, C!2) | in(B!3, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3))))),
% 0.24/0.42 inference(transitivity,[status(thm)],[42, 40])).
% 0.24/0.42 tff(44,plain,
% 0.24/0.42 (((~![A: $i, B: $i, C: $i] : (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))) | (~((in(B!3, C!2) | in(A!4, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3))))) <=> ((~![A: $i, B: $i, C: $i] : (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))) | (~((in(A!4, C!2) | in(B!3, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3)))))),
% 0.24/0.42 inference(monotonicity,[status(thm)],[43])).
% 0.24/0.42 tff(45,plain,
% 0.24/0.42 (((~![A: $i, B: $i, C: $i] : (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))) | (~((in(B!3, C!2) | in(A!4, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3))))) <=> ((~![A: $i, B: $i, C: $i] : (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))) | (~((in(A!4, C!2) | in(B!3, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3)))))),
% 0.24/0.42 inference(transitivity,[status(thm)],[44, 39])).
% 0.24/0.42 tff(46,plain,
% 0.24/0.42 ((~![A: $i, B: $i, C: $i] : (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))) | (~((in(B!3, C!2) | in(A!4, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3))))),
% 0.24/0.42 inference(quant_inst,[status(thm)],[])).
% 0.24/0.42 tff(47,plain,
% 0.24/0.42 ((~![A: $i, B: $i, C: $i] : (~((in(B, C) | in(A, C)) <=> (set_difference(unordered_pair(A, B), C) = unordered_pair(A, B))))) | (~((in(A!4, C!2) | in(B!3, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3))))),
% 0.24/0.42 inference(modus_ponens,[status(thm)],[46, 45])).
% 0.24/0.42 tff(48,plain,
% 0.24/0.42 (~((in(A!4, C!2) | in(B!3, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3)))),
% 0.24/0.42 inference(unit_resolution,[status(thm)],[47, 38])).
% 0.24/0.42 tff(49,plain,
% 0.24/0.42 (~(set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3))),
% 0.24/0.42 inference(and_elim,[status(thm)],[21])).
% 0.24/0.42 tff(50,plain,
% 0.24/0.42 (((in(A!4, C!2) | in(B!3, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3))) | (in(A!4, C!2) | in(B!3, C!2)) | (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3))),
% 0.24/0.42 inference(tautology,[status(thm)],[])).
% 0.24/0.42 tff(51,plain,
% 0.24/0.42 (((in(A!4, C!2) | in(B!3, C!2)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = unordered_pair(A!4, B!3))) | (in(A!4, C!2) | in(B!3, C!2))),
% 0.24/0.42 inference(unit_resolution,[status(thm)],[50, 49])).
% 0.24/0.42 tff(52,plain,
% 0.24/0.42 (in(A!4, C!2) | in(B!3, C!2)),
% 0.24/0.42 inference(unit_resolution,[status(thm)],[51, 48])).
% 0.24/0.42 tff(53,plain,
% 0.24/0.42 ((~(in(A!4, C!2) | in(B!3, C!2))) | in(A!4, C!2) | in(B!3, C!2)),
% 0.24/0.42 inference(tautology,[status(thm)],[])).
% 0.24/0.42 tff(54,plain,
% 0.24/0.42 (in(A!4, C!2) | in(B!3, C!2)),
% 0.24/0.42 inference(unit_resolution,[status(thm)],[53, 52])).
% 0.24/0.42 tff(55,plain,
% 0.24/0.42 (in(B!3, C!2)),
% 0.24/0.42 inference(unit_resolution,[status(thm)],[54, 1])).
% 0.24/0.42 tff(56,plain,
% 0.24/0.42 ((in(B!3, C!2) | (A!4 = B!3)) | (~in(B!3, C!2))),
% 0.24/0.42 inference(tautology,[status(thm)],[])).
% 0.24/0.42 tff(57,plain,
% 0.24/0.42 ($false),
% 0.24/0.42 inference(unit_resolution,[status(thm)],[56, 55, 28])).
% 0.24/0.42 tff(58,plain,(in(A!4, C!2)), inference(lemma,lemma(discharge,[]))).
% 0.24/0.42 tff(59,plain,
% 0.24/0.42 (^[A: $i, B: $i, C: $i] : refl((~(((~in(A, C)) | (~in(B, C))) <=> (set_difference(unordered_pair(A, B), C) = empty_set))) <=> (~(((~in(A, C)) | (~in(B, C))) <=> (set_difference(unordered_pair(A, B), C) = empty_set))))),
% 0.24/0.42 inference(bind,[status(th)],[])).
% 0.24/0.42 tff(60,plain,
% 0.24/0.42 (![A: $i, B: $i, C: $i] : (~(((~in(A, C)) | (~in(B, C))) <=> (set_difference(unordered_pair(A, B), C) = empty_set))) <=> ![A: $i, B: $i, C: $i] : (~(((~in(A, C)) | (~in(B, C))) <=> (set_difference(unordered_pair(A, B), C) = empty_set)))),
% 0.24/0.42 inference(quant_intro,[status(thm)],[59])).
% 0.24/0.42 tff(61,plain,
% 0.24/0.42 (^[A: $i, B: $i, C: $i] : trans(trans(monotonicity(rewrite((in(A, C) & in(B, C)) <=> (~((~in(A, C)) | (~in(B, C))))), (((set_difference(unordered_pair(A, B), C) = empty_set) <=> (in(A, C) & in(B, C))) <=> ((set_difference(unordered_pair(A, B), C) = empty_set) <=> (~((~in(A, C)) | (~in(B, C))))))), rewrite(((set_difference(unordered_pair(A, B), C) = empty_set) <=> (~((~in(A, C)) | (~in(B, C))))) <=> (~(((~in(A, C)) | (~in(B, C))) <=> (set_difference(unordered_pair(A, B), C) = empty_set)))), (((set_difference(unordered_pair(A, B), C) = empty_set) <=> (in(A, C) & in(B, C))) <=> (~(((~in(A, C)) | (~in(B, C))) <=> (set_difference(unordered_pair(A, B), C) = empty_set))))), rewrite((~(((~in(A, C)) | (~in(B, C))) <=> (set_difference(unordered_pair(A, B), C) = empty_set))) <=> (~(((~in(A, C)) | (~in(B, C))) <=> (set_difference(unordered_pair(A, B), C) = empty_set)))), (((set_difference(unordered_pair(A, B), C) = empty_set) <=> (in(A, C) & in(B, C))) <=> (~(((~in(A, C)) | (~in(B, C))) <=> (set_difference(unordered_pair(A, B), C) = empty_set)))))),
% 0.24/0.42 inference(bind,[status(th)],[])).
% 0.24/0.42 tff(62,plain,
% 0.24/0.42 (![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = empty_set) <=> (in(A, C) & in(B, C))) <=> ![A: $i, B: $i, C: $i] : (~(((~in(A, C)) | (~in(B, C))) <=> (set_difference(unordered_pair(A, B), C) = empty_set)))),
% 0.24/0.42 inference(quant_intro,[status(thm)],[61])).
% 0.24/0.42 tff(63,plain,
% 0.24/0.42 (![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = empty_set) <=> (in(A, C) & in(B, C))) <=> ![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = empty_set) <=> (in(A, C) & in(B, C)))),
% 0.24/0.42 inference(rewrite,[status(thm)],[])).
% 0.24/0.42 tff(64,axiom,(![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = empty_set) <=> (in(A, C) & in(B, C)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t73_zfmisc_1')).
% 0.24/0.42 tff(65,plain,
% 0.24/0.42 (![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = empty_set) <=> (in(A, C) & in(B, C)))),
% 0.24/0.42 inference(modus_ponens,[status(thm)],[64, 63])).
% 0.24/0.42 tff(66,plain,(
% 0.24/0.42 ![A: $i, B: $i, C: $i] : ((set_difference(unordered_pair(A, B), C) = empty_set) <=> (in(A, C) & in(B, C)))),
% 0.24/0.42 inference(skolemize,[status(sab)],[65])).
% 0.24/0.42 tff(67,plain,
% 0.24/0.42 (![A: $i, B: $i, C: $i] : (~(((~in(A, C)) | (~in(B, C))) <=> (set_difference(unordered_pair(A, B), C) = empty_set)))),
% 0.24/0.42 inference(modus_ponens,[status(thm)],[66, 62])).
% 0.24/0.42 tff(68,plain,
% 0.24/0.42 (![A: $i, B: $i, C: $i] : (~(((~in(A, C)) | (~in(B, C))) <=> (set_difference(unordered_pair(A, B), C) = empty_set)))),
% 0.24/0.42 inference(modus_ponens,[status(thm)],[67, 60])).
% 0.24/0.42 tff(69,plain,
% 0.24/0.42 ((~![A: $i, B: $i, C: $i] : (~(((~in(A, C)) | (~in(B, C))) <=> (set_difference(unordered_pair(A, B), C) = empty_set)))) | (~(((~in(A!4, C!2)) | (~in(B!3, C!2))) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = empty_set)))),
% 0.24/0.42 inference(quant_inst,[status(thm)],[])).
% 0.24/0.42 tff(70,plain,
% 0.24/0.42 (~(((~in(A!4, C!2)) | (~in(B!3, C!2))) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = empty_set))),
% 0.24/0.42 inference(unit_resolution,[status(thm)],[69, 68])).
% 0.24/0.42 tff(71,plain,
% 0.24/0.42 (~(set_difference(unordered_pair(A!4, B!3), C!2) = empty_set)),
% 0.24/0.42 inference(and_elim,[status(thm)],[21])).
% 0.24/0.42 tff(72,plain,
% 0.24/0.42 ((((~in(A!4, C!2)) | (~in(B!3, C!2))) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = empty_set)) | ((~in(A!4, C!2)) | (~in(B!3, C!2))) | (set_difference(unordered_pair(A!4, B!3), C!2) = empty_set)),
% 0.24/0.42 inference(tautology,[status(thm)],[])).
% 0.24/0.42 tff(73,plain,
% 0.24/0.42 ((((~in(A!4, C!2)) | (~in(B!3, C!2))) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = empty_set)) | ((~in(A!4, C!2)) | (~in(B!3, C!2)))),
% 0.24/0.42 inference(unit_resolution,[status(thm)],[72, 71])).
% 0.24/0.42 tff(74,plain,
% 0.24/0.42 ((~in(A!4, C!2)) | (~in(B!3, C!2))),
% 0.24/0.42 inference(unit_resolution,[status(thm)],[73, 70])).
% 0.24/0.42 tff(75,plain,
% 0.24/0.42 ((~((~in(A!4, C!2)) | (~in(B!3, C!2)))) | (~in(A!4, C!2)) | (~in(B!3, C!2))),
% 0.24/0.42 inference(tautology,[status(thm)],[])).
% 0.24/0.42 tff(76,plain,
% 0.24/0.42 ((~in(A!4, C!2)) | (~in(B!3, C!2))),
% 0.24/0.42 inference(unit_resolution,[status(thm)],[75, 74])).
% 0.24/0.42 tff(77,plain,
% 0.24/0.42 (~in(B!3, C!2)),
% 0.24/0.42 inference(unit_resolution,[status(thm)],[76, 58])).
% 0.24/0.42 tff(78,plain,
% 0.24/0.42 ((in(A!4, C!2) | (B!3 = A!4)) | (~in(A!4, C!2))),
% 0.24/0.42 inference(tautology,[status(thm)],[])).
% 0.24/0.42 tff(79,plain,
% 0.24/0.42 (in(A!4, C!2) | (B!3 = A!4)),
% 0.24/0.42 inference(unit_resolution,[status(thm)],[78, 58])).
% 0.24/0.42 tff(80,plain,
% 0.24/0.42 ((~![A: $i, B: $i, C: $i] : (~((in(A, C) | (~(in(B, C) | (A = B)))) <=> (set_difference(unordered_pair(A, B), C) = singleton(A))))) | (~((in(B!3, C!2) | (~(in(A!4, C!2) | (B!3 = A!4)))) <=> (set_difference(unordered_pair(B!3, A!4), C!2) = singleton(B!3))))),
% 0.24/0.42 inference(quant_inst,[status(thm)],[])).
% 0.24/0.42 tff(81,plain,
% 0.24/0.42 (~((in(B!3, C!2) | (~(in(A!4, C!2) | (B!3 = A!4)))) <=> (set_difference(unordered_pair(B!3, A!4), C!2) = singleton(B!3)))),
% 0.24/0.42 inference(unit_resolution,[status(thm)],[80, 11])).
% 0.24/0.42 tff(82,plain,
% 0.24/0.42 (^[A: $i, B: $i] : refl((unordered_pair(A, B) = unordered_pair(B, A)) <=> (unordered_pair(A, B) = unordered_pair(B, A)))),
% 0.24/0.42 inference(bind,[status(th)],[])).
% 0.24/0.42 tff(83,plain,
% 0.24/0.42 (![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A)) <=> ![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))),
% 0.24/0.42 inference(quant_intro,[status(thm)],[82])).
% 0.24/0.42 tff(84,plain,
% 0.24/0.42 (![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A)) <=> ![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))),
% 0.24/0.42 inference(rewrite,[status(thm)],[])).
% 0.24/0.42 tff(85,axiom,(![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','commutativity_k2_tarski')).
% 0.24/0.42 tff(86,plain,
% 0.24/0.42 (![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))),
% 0.24/0.42 inference(modus_ponens,[status(thm)],[85, 84])).
% 0.24/0.42 tff(87,plain,(
% 0.24/0.42 ![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))),
% 0.24/0.42 inference(skolemize,[status(sab)],[86])).
% 0.24/0.42 tff(88,plain,
% 0.24/0.42 (![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))),
% 0.24/0.42 inference(modus_ponens,[status(thm)],[87, 83])).
% 0.24/0.42 tff(89,plain,
% 0.24/0.42 ((~![A: $i, B: $i] : (unordered_pair(A, B) = unordered_pair(B, A))) | (unordered_pair(A!4, B!3) = unordered_pair(B!3, A!4))),
% 0.24/0.43 inference(quant_inst,[status(thm)],[])).
% 0.24/0.43 tff(90,plain,
% 0.24/0.43 (unordered_pair(A!4, B!3) = unordered_pair(B!3, A!4)),
% 0.24/0.43 inference(unit_resolution,[status(thm)],[89, 88])).
% 0.24/0.43 tff(91,plain,
% 0.24/0.43 (unordered_pair(B!3, A!4) = unordered_pair(A!4, B!3)),
% 0.24/0.43 inference(symmetry,[status(thm)],[90])).
% 0.24/0.43 tff(92,plain,
% 0.24/0.43 (set_difference(unordered_pair(B!3, A!4), C!2) = set_difference(unordered_pair(A!4, B!3), C!2)),
% 0.24/0.43 inference(monotonicity,[status(thm)],[91])).
% 0.24/0.43 tff(93,plain,
% 0.24/0.43 ((set_difference(unordered_pair(B!3, A!4), C!2) = singleton(B!3)) <=> (set_difference(unordered_pair(A!4, B!3), C!2) = singleton(B!3))),
% 0.24/0.43 inference(monotonicity,[status(thm)],[92])).
% 0.24/0.43 tff(94,plain,
% 0.24/0.43 ((set_difference(unordered_pair(A!4, B!3), C!2) = singleton(B!3)) <=> (set_difference(unordered_pair(B!3, A!4), C!2) = singleton(B!3))),
% 0.24/0.43 inference(symmetry,[status(thm)],[93])).
% 0.24/0.43 tff(95,plain,
% 0.24/0.43 ((~(set_difference(unordered_pair(A!4, B!3), C!2) = singleton(B!3))) <=> (~(set_difference(unordered_pair(B!3, A!4), C!2) = singleton(B!3)))),
% 0.24/0.43 inference(monotonicity,[status(thm)],[94])).
% 0.24/0.43 tff(96,plain,
% 0.24/0.43 (~(set_difference(unordered_pair(A!4, B!3), C!2) = singleton(B!3))),
% 0.24/0.43 inference(and_elim,[status(thm)],[21])).
% 0.24/0.43 tff(97,plain,
% 0.24/0.43 (~(set_difference(unordered_pair(B!3, A!4), C!2) = singleton(B!3))),
% 0.24/0.43 inference(modus_ponens,[status(thm)],[96, 95])).
% 0.24/0.43 tff(98,plain,
% 0.24/0.43 (((in(B!3, C!2) | (~(in(A!4, C!2) | (B!3 = A!4)))) <=> (set_difference(unordered_pair(B!3, A!4), C!2) = singleton(B!3))) | (in(B!3, C!2) | (~(in(A!4, C!2) | (B!3 = A!4)))) | (set_difference(unordered_pair(B!3, A!4), C!2) = singleton(B!3))),
% 0.24/0.43 inference(tautology,[status(thm)],[])).
% 0.24/0.43 tff(99,plain,
% 0.24/0.43 (in(B!3, C!2) | (~(in(A!4, C!2) | (B!3 = A!4)))),
% 0.24/0.43 inference(unit_resolution,[status(thm)],[98, 97, 81])).
% 0.24/0.43 tff(100,plain,
% 0.24/0.43 ((~(in(B!3, C!2) | (~(in(A!4, C!2) | (B!3 = A!4))))) | in(B!3, C!2) | (~(in(A!4, C!2) | (B!3 = A!4)))),
% 0.24/0.43 inference(tautology,[status(thm)],[])).
% 0.24/0.43 tff(101,plain,
% 0.24/0.43 (in(B!3, C!2) | (~(in(A!4, C!2) | (B!3 = A!4)))),
% 0.24/0.43 inference(unit_resolution,[status(thm)],[100, 99])).
% 0.24/0.43 tff(102,plain,
% 0.24/0.43 ($false),
% 0.24/0.43 inference(unit_resolution,[status(thm)],[101, 79, 77])).
% 0.24/0.43 % SZS output end Proof
% 0.24/0.43 % E exiting
%------------------------------------------------------------------------------