%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET985+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n013.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 05:26:45 AM UTC 2025
% Result : Theorem 0.13s 0.38s
% Output : Proof 0.13s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SET985+1 : TPTP v9.0.0. Released v3.2.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n013.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Jun 20 05:54:53 EDT 2025
% 0.13/0.35 % CPUTime :
% 0.13/0.38 % SZS status Theorem
% 0.13/0.38 % SZS output start Proof
% 0.13/0.38 tff(subset_type, type, (
% 0.13/0.38 subset: ( $i * $i ) > $o)).
% 0.13/0.38 tff(tptp_fun_D_3_type, type, (
% 0.13/0.38 tptp_fun_D_3: $i)).
% 0.13/0.38 tff(tptp_fun_B_5_type, type, (
% 0.13/0.38 tptp_fun_B_5: $i)).
% 0.13/0.38 tff(tptp_fun_C_4_type, type, (
% 0.13/0.38 tptp_fun_C_4: $i)).
% 0.13/0.38 tff(tptp_fun_A_2_type, type, (
% 0.13/0.38 tptp_fun_A_2: $i)).
% 0.13/0.38 tff(cartesian_product2_type, type, (
% 0.13/0.38 cartesian_product2: ( $i * $i ) > $i)).
% 0.13/0.38 tff(empty_type, type, (
% 0.13/0.38 empty: $i > $o)).
% 0.13/0.38 tff(empty_set_type, type, (
% 0.13/0.38 empty_set: $i)).
% 0.13/0.38 tff(1,plain,
% 0.13/0.38 ((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) <=> (~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))))),
% 0.13/0.38 inference(rewrite,[status(thm)],[])).
% 0.13/0.38 tff(2,plain,
% 0.13/0.38 (((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3)) <=> ((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3))),
% 0.13/0.38 inference(monotonicity,[status(thm)],[1])).
% 0.13/0.38 tff(3,plain,
% 0.13/0.38 ((~((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3))) <=> (~((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3)))),
% 0.13/0.38 inference(monotonicity,[status(thm)],[2])).
% 0.13/0.38 tff(4,plain,
% 0.13/0.38 (((~empty(A!2)) & (~((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3)))) <=> ((~empty(A!2)) & (~((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3))))),
% 0.13/0.38 inference(monotonicity,[status(thm)],[3])).
% 0.13/0.38 tff(5,plain,
% 0.13/0.38 ((~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D)))) <=> (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D))))),
% 0.13/0.38 inference(rewrite,[status(thm)],[])).
% 0.13/0.38 tff(6,plain,
% 0.13/0.38 ((~![A: $i] : ((~empty(A)) => ![B: $i, C: $i, D: $i] : ((subset(cartesian_product2(A, B), cartesian_product2(C, D)) | subset(cartesian_product2(B, A), cartesian_product2(D, C))) => subset(B, D)))) <=> (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D))))),
% 0.13/0.38 inference(rewrite,[status(thm)],[])).
% 0.13/0.38 tff(7,axiom,(~![A: $i] : ((~empty(A)) => ![B: $i, C: $i, D: $i] : ((subset(cartesian_product2(A, B), cartesian_product2(C, D)) | subset(cartesian_product2(B, A), cartesian_product2(D, C))) => subset(B, D)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t139_zfmisc_1')).
% 0.13/0.38 tff(8,plain,
% 0.13/0.38 (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D)))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[7, 6])).
% 0.13/0.38 tff(9,plain,
% 0.13/0.38 (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D)))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[8, 5])).
% 0.13/0.38 tff(10,plain,
% 0.13/0.38 (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D)))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[9, 5])).
% 0.13/0.38 tff(11,plain,
% 0.13/0.38 (~![A: $i] : (empty(A) | ![B: $i, C: $i, D: $i] : ((~(subset(cartesian_product2(B, A), cartesian_product2(D, C)) | subset(cartesian_product2(A, B), cartesian_product2(C, D)))) | subset(B, D)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[10, 5])).
% 0.13/0.39 tff(12,plain,
% 0.13/0.39 ((~empty(A!2)) & (~((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[11, 4])).
% 0.13/0.39 tff(13,plain,
% 0.13/0.39 (~((~(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3)))) | subset(B!5, D!3))),
% 0.13/0.39 inference(and_elim,[status(thm)],[12])).
% 0.13/0.39 tff(14,plain,
% 0.13/0.39 (~subset(B!5, D!3)),
% 0.13/0.39 inference(or_elim,[status(thm)],[13])).
% 0.13/0.39 tff(15,plain,
% 0.13/0.39 (((~subset(A!2, C!4)) | (~subset(B!5, D!3))) | subset(B!5, D!3)),
% 0.13/0.39 inference(tautology,[status(thm)],[])).
% 0.13/0.39 tff(16,plain,
% 0.13/0.39 ((~subset(A!2, C!4)) | (~subset(B!5, D!3))),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[15, 14])).
% 0.13/0.39 tff(17,assumption,(~((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))), introduced(assumption)).
% 0.13/0.39 tff(18,plain,
% 0.13/0.39 (^[A: $i, B: $i] : refl(((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set))) <=> ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set))))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(19,plain,
% 0.13/0.39 (![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set))) <=> ![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[18])).
% 0.13/0.39 tff(20,plain,
% 0.13/0.39 (![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set))) <=> ![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(21,plain,
% 0.13/0.39 (^[A: $i, B: $i] : rewrite(((cartesian_product2(A, B) = empty_set) <=> ((A = empty_set) | (B = empty_set))) <=> ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set))))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(22,plain,
% 0.13/0.39 (![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((A = empty_set) | (B = empty_set))) <=> ![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[21])).
% 0.13/0.39 tff(23,axiom,(![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((A = empty_set) | (B = empty_set)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t113_zfmisc_1')).
% 0.13/0.39 tff(24,plain,
% 0.13/0.39 (![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[23, 22])).
% 0.13/0.39 tff(25,plain,
% 0.13/0.39 (![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[24, 20])).
% 0.13/0.39 tff(26,plain,(
% 0.13/0.39 ![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.39 inference(skolemize,[status(sab)],[25])).
% 0.13/0.39 tff(27,plain,
% 0.13/0.39 (![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[26, 19])).
% 0.13/0.39 tff(28,plain,
% 0.13/0.39 (((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))) <=> ((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set))))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(29,plain,
% 0.13/0.39 (((cartesian_product2(A!2, B!5) = empty_set) <=> ((B!5 = empty_set) | (A!2 = empty_set))) <=> ((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(30,plain,
% 0.13/0.39 (((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((B!5 = empty_set) | (A!2 = empty_set)))) <=> ((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set))))),
% 0.13/0.39 inference(monotonicity,[status(thm)],[29])).
% 0.13/0.39 tff(31,plain,
% 0.13/0.39 (((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((B!5 = empty_set) | (A!2 = empty_set)))) <=> ((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set))))),
% 0.13/0.39 inference(transitivity,[status(thm)],[30, 28])).
% 0.13/0.39 tff(32,plain,
% 0.13/0.39 ((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((B!5 = empty_set) | (A!2 = empty_set)))),
% 0.13/0.39 inference(quant_inst,[status(thm)],[])).
% 0.13/0.39 tff(33,plain,
% 0.13/0.39 ((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[32, 31])).
% 0.13/0.39 tff(34,plain,
% 0.13/0.39 ($false),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[33, 27, 17])).
% 0.13/0.39 tff(35,plain,((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set))), inference(lemma,lemma(discharge,[]))).
% 0.13/0.39 tff(36,assumption,(B!5 = empty_set), introduced(assumption)).
% 0.13/0.39 tff(37,plain,
% 0.13/0.39 (empty_set = B!5),
% 0.13/0.39 inference(symmetry,[status(thm)],[36])).
% 0.13/0.39 tff(38,plain,
% 0.13/0.39 (subset(empty_set, D!3) <=> subset(B!5, D!3)),
% 0.13/0.39 inference(monotonicity,[status(thm)],[37])).
% 0.13/0.39 tff(39,plain,
% 0.13/0.39 (subset(B!5, D!3) <=> subset(empty_set, D!3)),
% 0.13/0.39 inference(symmetry,[status(thm)],[38])).
% 0.13/0.39 tff(40,plain,
% 0.13/0.39 ((~subset(B!5, D!3)) <=> (~subset(empty_set, D!3))),
% 0.13/0.39 inference(monotonicity,[status(thm)],[39])).
% 0.13/0.39 tff(41,plain,
% 0.13/0.39 (~subset(empty_set, D!3)),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[14, 40])).
% 0.13/0.39 tff(42,plain,
% 0.13/0.39 (^[A: $i] : refl(subset(empty_set, A) <=> subset(empty_set, A))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(43,plain,
% 0.13/0.39 (![A: $i] : subset(empty_set, A) <=> ![A: $i] : subset(empty_set, A)),
% 0.13/0.39 inference(quant_intro,[status(thm)],[42])).
% 0.13/0.39 tff(44,plain,
% 0.13/0.39 (![A: $i] : subset(empty_set, A) <=> ![A: $i] : subset(empty_set, A)),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(45,axiom,(![A: $i] : subset(empty_set, A)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t2_xboole_1')).
% 0.13/0.39 tff(46,plain,
% 0.13/0.39 (![A: $i] : subset(empty_set, A)),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[45, 44])).
% 0.13/0.39 tff(47,plain,(
% 0.13/0.39 ![A: $i] : subset(empty_set, A)),
% 0.13/0.39 inference(skolemize,[status(sab)],[46])).
% 0.13/0.39 tff(48,plain,
% 0.13/0.39 (![A: $i] : subset(empty_set, A)),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[47, 43])).
% 0.13/0.39 tff(49,plain,
% 0.13/0.39 ((~![A: $i] : subset(empty_set, A)) | subset(empty_set, D!3)),
% 0.13/0.39 inference(quant_inst,[status(thm)],[])).
% 0.13/0.39 tff(50,plain,
% 0.13/0.39 (subset(empty_set, D!3)),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[49, 48])).
% 0.13/0.39 tff(51,plain,
% 0.13/0.39 ($false),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[50, 41])).
% 0.13/0.39 tff(52,plain,(~(B!5 = empty_set)), inference(lemma,lemma(discharge,[]))).
% 0.13/0.39 tff(53,assumption,(A!2 = empty_set), introduced(assumption)).
% 0.13/0.39 tff(54,plain,
% 0.13/0.39 (empty(A!2) <=> empty(empty_set)),
% 0.13/0.39 inference(monotonicity,[status(thm)],[53])).
% 0.13/0.39 tff(55,plain,
% 0.13/0.39 (empty(empty_set) <=> empty(A!2)),
% 0.13/0.39 inference(symmetry,[status(thm)],[54])).
% 0.13/0.39 tff(56,plain,
% 0.13/0.39 (empty(empty_set) <=> empty(empty_set)),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(57,axiom,(empty(empty_set)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','fc1_xboole_0')).
% 0.13/0.39 tff(58,plain,
% 0.13/0.39 (empty(empty_set)),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[57, 56])).
% 0.13/0.39 tff(59,plain,
% 0.13/0.39 (empty(A!2)),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[58, 55])).
% 0.13/0.39 tff(60,plain,
% 0.13/0.39 (~empty(A!2)),
% 0.13/0.39 inference(and_elim,[status(thm)],[12])).
% 0.13/0.39 tff(61,plain,
% 0.13/0.39 ($false),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[60, 59])).
% 0.13/0.39 tff(62,plain,(~(A!2 = empty_set)), inference(lemma,lemma(discharge,[]))).
% 0.13/0.39 tff(63,plain,
% 0.13/0.39 ((~((A!2 = empty_set) | (B!5 = empty_set))) | (A!2 = empty_set) | (B!5 = empty_set)),
% 0.13/0.39 inference(tautology,[status(thm)],[])).
% 0.13/0.39 tff(64,plain,
% 0.13/0.39 (~((A!2 = empty_set) | (B!5 = empty_set))),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[63, 62, 52])).
% 0.13/0.39 tff(65,plain,
% 0.13/0.39 ((~((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))) | (~(cartesian_product2(A!2, B!5) = empty_set)) | ((A!2 = empty_set) | (B!5 = empty_set))),
% 0.13/0.39 inference(tautology,[status(thm)],[])).
% 0.13/0.39 tff(66,plain,
% 0.13/0.39 ((~((cartesian_product2(A!2, B!5) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))) | (~(cartesian_product2(A!2, B!5) = empty_set))),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[65, 64])).
% 0.13/0.39 tff(67,plain,
% 0.13/0.39 (~(cartesian_product2(A!2, B!5) = empty_set)),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[66, 35])).
% 0.13/0.39 tff(68,plain,
% 0.13/0.39 ((~![A: $i, B: $i] : ((cartesian_product2(A, B) = empty_set) <=> ((B = empty_set) | (A = empty_set)))) | ((cartesian_product2(B!5, A!2) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))),
% 0.13/0.39 inference(quant_inst,[status(thm)],[])).
% 0.13/0.39 tff(69,plain,
% 0.13/0.39 ((cartesian_product2(B!5, A!2) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set))),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[68, 27])).
% 0.13/0.39 tff(70,plain,
% 0.13/0.39 (((~subset(B!5, D!3)) | (~subset(A!2, C!4))) | subset(B!5, D!3)),
% 0.13/0.39 inference(tautology,[status(thm)],[])).
% 0.13/0.39 tff(71,plain,
% 0.13/0.39 ((~subset(B!5, D!3)) | (~subset(A!2, C!4))),
% 0.13/0.39 inference(unit_resolution,[status(thm)],[70, 14])).
% 0.13/0.39 tff(72,assumption,(subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4))), introduced(assumption)).
% 0.13/0.39 tff(73,plain,
% 0.13/0.39 (^[A: $i, B: $i, C: $i, D: $i] : refl(((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D))))) <=> ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D))))))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(74,plain,
% 0.13/0.39 (![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D))))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[73])).
% 0.13/0.39 tff(75,plain,
% 0.13/0.39 (^[A: $i, B: $i, C: $i, D: $i] : trans(monotonicity(rewrite((subset(A, C) & subset(B, D)) <=> (~((~subset(A, C)) | (~subset(B, D))))), (((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))) <=> ((~((~subset(A, C)) | (~subset(B, D)))) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))))), rewrite(((~((~subset(A, C)) | (~subset(B, D)))) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))) <=> ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))), (((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))) <=> ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(76,plain,
% 0.13/0.39 (![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[75])).
% 0.13/0.39 tff(77,plain,
% 0.13/0.39 (![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(78,plain,
% 0.13/0.39 (^[A: $i, B: $i, C: $i, D: $i] : rewrite((subset(cartesian_product2(A, B), cartesian_product2(C, D)) => ((cartesian_product2(A, B) = empty_set) | (subset(A, C) & subset(B, D)))) <=> ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D)))))),
% 0.13/0.39 inference(bind,[status(th)],[])).
% 0.13/0.39 tff(79,plain,
% 0.13/0.39 (![A: $i, B: $i, C: $i, D: $i] : (subset(cartesian_product2(A, B), cartesian_product2(C, D)) => ((cartesian_product2(A, B) = empty_set) | (subset(A, C) & subset(B, D)))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))))),
% 0.13/0.39 inference(quant_intro,[status(thm)],[78])).
% 0.13/0.39 tff(80,axiom,(![A: $i, B: $i, C: $i, D: $i] : (subset(cartesian_product2(A, B), cartesian_product2(C, D)) => ((cartesian_product2(A, B) = empty_set) | (subset(A, C) & subset(B, D))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t138_zfmisc_1')).
% 0.13/0.39 tff(81,plain,
% 0.13/0.39 (![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[80, 79])).
% 0.13/0.39 tff(82,plain,
% 0.13/0.39 (![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[81, 77])).
% 0.13/0.39 tff(83,plain,(
% 0.13/0.39 ![A: $i, B: $i, C: $i, D: $i] : ((subset(A, C) & subset(B, D)) | (cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))))),
% 0.13/0.39 inference(skolemize,[status(sab)],[82])).
% 0.13/0.39 tff(84,plain,
% 0.13/0.39 (![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[83, 76])).
% 0.13/0.39 tff(85,plain,
% 0.13/0.39 (![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))),
% 0.13/0.39 inference(modus_ponens,[status(thm)],[84, 74])).
% 0.13/0.39 tff(86,plain,
% 0.13/0.39 (((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | ((cartesian_product2(B!5, A!2) = empty_set) | (~subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4))) | (~((~subset(B!5, D!3)) | (~subset(A!2, C!4)))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | (cartesian_product2(B!5, A!2) = empty_set) | (~subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4))) | (~((~subset(B!5, D!3)) | (~subset(A!2, C!4)))))),
% 0.13/0.39 inference(rewrite,[status(thm)],[])).
% 0.13/0.39 tff(87,plain,
% 0.13/0.39 ((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | ((cartesian_product2(B!5, A!2) = empty_set) | (~subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4))) | (~((~subset(B!5, D!3)) | (~subset(A!2, C!4)))))),
% 0.13/0.39 inference(quant_inst,[status(thm)],[])).
% 0.13/0.39 tff(88,plain,
% 0.13/0.39 ((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | (cartesian_product2(B!5, A!2) = empty_set) | (~subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4))) | (~((~subset(B!5, D!3)) | (~subset(A!2, C!4))))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[87, 86])).
% 0.13/0.40 tff(89,plain,
% 0.13/0.40 (cartesian_product2(B!5, A!2) = empty_set),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[88, 85, 72, 71])).
% 0.13/0.40 tff(90,plain,
% 0.13/0.40 ((~((cartesian_product2(B!5, A!2) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))) | (~(cartesian_product2(B!5, A!2) = empty_set)) | ((A!2 = empty_set) | (B!5 = empty_set))),
% 0.13/0.40 inference(tautology,[status(thm)],[])).
% 0.13/0.40 tff(91,plain,
% 0.13/0.40 ((~((cartesian_product2(B!5, A!2) = empty_set) <=> ((A!2 = empty_set) | (B!5 = empty_set)))) | (~(cartesian_product2(B!5, A!2) = empty_set))),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[90, 64])).
% 0.13/0.40 tff(92,plain,
% 0.13/0.40 ($false),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[91, 89, 69])).
% 0.13/0.40 tff(93,plain,(~subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4))), inference(lemma,lemma(discharge,[]))).
% 0.13/0.40 tff(94,plain,
% 0.13/0.40 (subset(cartesian_product2(B!5, A!2), cartesian_product2(D!3, C!4)) | subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))),
% 0.13/0.40 inference(or_elim,[status(thm)],[13])).
% 0.13/0.40 tff(95,plain,
% 0.13/0.40 (subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[94, 93])).
% 0.13/0.40 tff(96,plain,
% 0.13/0.40 (((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | ((cartesian_product2(A!2, B!5) = empty_set) | (~subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))) | (~((~subset(A!2, C!4)) | (~subset(B!5, D!3)))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | (cartesian_product2(A!2, B!5) = empty_set) | (~subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))) | (~((~subset(A!2, C!4)) | (~subset(B!5, D!3)))))),
% 0.13/0.40 inference(rewrite,[status(thm)],[])).
% 0.13/0.40 tff(97,plain,
% 0.13/0.40 ((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | ((cartesian_product2(A!2, B!5) = empty_set) | (~subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))) | (~((~subset(A!2, C!4)) | (~subset(B!5, D!3)))))),
% 0.13/0.40 inference(quant_inst,[status(thm)],[])).
% 0.13/0.40 tff(98,plain,
% 0.13/0.40 ((~![A: $i, B: $i, C: $i, D: $i] : ((cartesian_product2(A, B) = empty_set) | (~subset(cartesian_product2(A, B), cartesian_product2(C, D))) | (~((~subset(A, C)) | (~subset(B, D)))))) | (cartesian_product2(A!2, B!5) = empty_set) | (~subset(cartesian_product2(A!2, B!5), cartesian_product2(C!4, D!3))) | (~((~subset(A!2, C!4)) | (~subset(B!5, D!3))))),
% 0.13/0.40 inference(modus_ponens,[status(thm)],[97, 96])).
% 0.13/0.40 tff(99,plain,
% 0.13/0.40 ($false),
% 0.13/0.40 inference(unit_resolution,[status(thm)],[98, 85, 95, 67, 16])).
% 0.13/0.40 % SZS output end Proof
% 0.13/0.40 % E exiting
%------------------------------------------------------------------------------