%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET978+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 0.37s
% Output : Proof 0.13s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : SET978+1 : TPTP v9.0.0. Released v3.2.0.
% 0.11/0.12 % Command : run_E %s %d THM
% 0.13/0.33 % Computer : n013.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Fri Jun 20 05:55:23 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.13/0.37 % SZS status Theorem
% 0.13/0.37 % SZS output start Proof
% 0.13/0.37 tff(disjoint_type, type, (
% 0.13/0.37 disjoint: ( $i * $i ) > $o)).
% 0.13/0.37 tff(singleton_type, type, (
% 0.13/0.37 singleton: $i > $i)).
% 0.13/0.37 tff(tptp_fun_B_4_type, type, (
% 0.13/0.37 tptp_fun_B_4: $i)).
% 0.13/0.37 tff(tptp_fun_A_5_type, type, (
% 0.13/0.37 tptp_fun_A_5: $i)).
% 0.13/0.37 tff(tptp_fun_D_2_type, type, (
% 0.13/0.37 tptp_fun_D_2: $i)).
% 0.13/0.37 tff(tptp_fun_C_3_type, type, (
% 0.13/0.37 tptp_fun_C_3: $i)).
% 0.13/0.37 tff(cartesian_product2_type, type, (
% 0.13/0.37 cartesian_product2: ( $i * $i ) > $i)).
% 0.13/0.37 tff(1,assumption,(~(disjoint(C!3, D!2) | disjoint(singleton(A!5), singleton(B!4)))), introduced(assumption)).
% 0.13/0.37 tff(2,plain,
% 0.13/0.37 (^[A: $i, B: $i] : refl(((A = B) | disjoint(singleton(A), singleton(B))) <=> ((A = B) | disjoint(singleton(A), singleton(B))))),
% 0.13/0.37 inference(bind,[status(th)],[])).
% 0.13/0.37 tff(3,plain,
% 0.13/0.37 (![A: $i, B: $i] : ((A = B) | disjoint(singleton(A), singleton(B))) <=> ![A: $i, B: $i] : ((A = B) | disjoint(singleton(A), singleton(B)))),
% 0.13/0.37 inference(quant_intro,[status(thm)],[2])).
% 0.13/0.37 tff(4,plain,
% 0.13/0.37 (![A: $i, B: $i] : ((A = B) | disjoint(singleton(A), singleton(B))) <=> ![A: $i, B: $i] : ((A = B) | disjoint(singleton(A), singleton(B)))),
% 0.13/0.37 inference(rewrite,[status(thm)],[])).
% 0.13/0.37 tff(5,plain,
% 0.13/0.37 (^[A: $i, B: $i] : rewrite(((~(A = B)) => disjoint(singleton(A), singleton(B))) <=> ((A = B) | disjoint(singleton(A), singleton(B))))),
% 0.13/0.37 inference(bind,[status(th)],[])).
% 0.13/0.37 tff(6,plain,
% 0.13/0.37 (![A: $i, B: $i] : ((~(A = B)) => disjoint(singleton(A), singleton(B))) <=> ![A: $i, B: $i] : ((A = B) | disjoint(singleton(A), singleton(B)))),
% 0.13/0.37 inference(quant_intro,[status(thm)],[5])).
% 0.13/0.37 tff(7,axiom,(![A: $i, B: $i] : ((~(A = B)) => disjoint(singleton(A), singleton(B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t17_zfmisc_1')).
% 0.13/0.37 tff(8,plain,
% 0.13/0.37 (![A: $i, B: $i] : ((A = B) | disjoint(singleton(A), singleton(B)))),
% 0.13/0.37 inference(modus_ponens,[status(thm)],[7, 6])).
% 0.13/0.37 tff(9,plain,
% 0.13/0.37 (![A: $i, B: $i] : ((A = B) | disjoint(singleton(A), singleton(B)))),
% 0.13/0.37 inference(modus_ponens,[status(thm)],[8, 4])).
% 0.13/0.37 tff(10,plain,(
% 0.13/0.37 ![A: $i, B: $i] : ((A = B) | disjoint(singleton(A), singleton(B)))),
% 0.13/0.37 inference(skolemize,[status(sab)],[9])).
% 0.13/0.37 tff(11,plain,
% 0.13/0.37 (![A: $i, B: $i] : ((A = B) | disjoint(singleton(A), singleton(B)))),
% 0.13/0.37 inference(modus_ponens,[status(thm)],[10, 3])).
% 0.13/0.37 tff(12,plain,
% 0.13/0.37 ((~![A: $i, B: $i, C: $i, D: $i] : ((A = B) | (disjoint(cartesian_product2(singleton(A), C), cartesian_product2(singleton(B), D)) & disjoint(cartesian_product2(C, singleton(A)), cartesian_product2(D, singleton(B)))))) <=> (~![A: $i, B: $i, C: $i, D: $i] : ((A = B) | (disjoint(cartesian_product2(singleton(A), C), cartesian_product2(singleton(B), D)) & disjoint(cartesian_product2(C, singleton(A)), cartesian_product2(D, singleton(B))))))),
% 0.13/0.37 inference(rewrite,[status(thm)],[])).
% 0.13/0.37 tff(13,plain,
% 0.13/0.37 ((~![A: $i, B: $i, C: $i, D: $i] : ((~(A = B)) => (disjoint(cartesian_product2(singleton(A), C), cartesian_product2(singleton(B), D)) & disjoint(cartesian_product2(C, singleton(A)), cartesian_product2(D, singleton(B)))))) <=> (~![A: $i, B: $i, C: $i, D: $i] : ((A = B) | (disjoint(cartesian_product2(singleton(A), C), cartesian_product2(singleton(B), D)) & disjoint(cartesian_product2(C, singleton(A)), cartesian_product2(D, singleton(B))))))),
% 0.13/0.37 inference(rewrite,[status(thm)],[])).
% 0.13/0.37 tff(14,axiom,(~![A: $i, B: $i, C: $i, D: $i] : ((~(A = B)) => (disjoint(cartesian_product2(singleton(A), C), cartesian_product2(singleton(B), D)) & disjoint(cartesian_product2(C, singleton(A)), cartesian_product2(D, singleton(B)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t131_zfmisc_1')).
% 0.13/0.37 tff(15,plain,
% 0.13/0.37 (~![A: $i, B: $i, C: $i, D: $i] : ((A = B) | (disjoint(cartesian_product2(singleton(A), C), cartesian_product2(singleton(B), D)) & disjoint(cartesian_product2(C, singleton(A)), cartesian_product2(D, singleton(B)))))),
% 0.13/0.37 inference(modus_ponens,[status(thm)],[14, 13])).
% 0.13/0.37 tff(16,plain,
% 0.13/0.37 (~![A: $i, B: $i, C: $i, D: $i] : ((A = B) | (disjoint(cartesian_product2(singleton(A), C), cartesian_product2(singleton(B), D)) & disjoint(cartesian_product2(C, singleton(A)), cartesian_product2(D, singleton(B)))))),
% 0.13/0.37 inference(modus_ponens,[status(thm)],[15, 12])).
% 0.13/0.37 tff(17,plain,
% 0.13/0.37 (~![A: $i, B: $i, C: $i, D: $i] : ((A = B) | (disjoint(cartesian_product2(singleton(A), C), cartesian_product2(singleton(B), D)) & disjoint(cartesian_product2(C, singleton(A)), cartesian_product2(D, singleton(B)))))),
% 0.13/0.37 inference(modus_ponens,[status(thm)],[16, 12])).
% 0.13/0.37 tff(18,plain,
% 0.13/0.37 (~![A: $i, B: $i, C: $i, D: $i] : ((A = B) | (disjoint(cartesian_product2(singleton(A), C), cartesian_product2(singleton(B), D)) & disjoint(cartesian_product2(C, singleton(A)), cartesian_product2(D, singleton(B)))))),
% 0.13/0.37 inference(modus_ponens,[status(thm)],[17, 12])).
% 0.13/0.37 tff(19,plain,(
% 0.13/0.37 ~((A!5 = B!4) | (disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2)) & disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4)))))),
% 0.13/0.37 inference(skolemize,[status(sab)],[18])).
% 0.13/0.37 tff(20,plain,
% 0.13/0.37 (~(A!5 = B!4)),
% 0.13/0.37 inference(or_elim,[status(thm)],[19])).
% 0.13/0.37 tff(21,plain,
% 0.13/0.37 (((~![A: $i, B: $i] : ((A = B) | disjoint(singleton(A), singleton(B)))) | ((A!5 = B!4) | disjoint(singleton(A!5), singleton(B!4)))) <=> ((~![A: $i, B: $i] : ((A = B) | disjoint(singleton(A), singleton(B)))) | (A!5 = B!4) | disjoint(singleton(A!5), singleton(B!4)))),
% 0.13/0.37 inference(rewrite,[status(thm)],[])).
% 0.13/0.37 tff(22,plain,
% 0.13/0.37 ((~![A: $i, B: $i] : ((A = B) | disjoint(singleton(A), singleton(B)))) | ((A!5 = B!4) | disjoint(singleton(A!5), singleton(B!4)))),
% 0.13/0.37 inference(quant_inst,[status(thm)],[])).
% 0.13/0.37 tff(23,plain,
% 0.13/0.37 ((~![A: $i, B: $i] : ((A = B) | disjoint(singleton(A), singleton(B)))) | (A!5 = B!4) | disjoint(singleton(A!5), singleton(B!4))),
% 0.13/0.37 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.13/0.37 tff(24,plain,
% 0.13/0.37 (disjoint(singleton(A!5), singleton(B!4))),
% 0.13/0.37 inference(unit_resolution,[status(thm)],[23, 20, 11])).
% 0.13/0.37 tff(25,plain,
% 0.13/0.37 ((disjoint(C!3, D!2) | disjoint(singleton(A!5), singleton(B!4))) | (~disjoint(singleton(A!5), singleton(B!4)))),
% 0.13/0.37 inference(tautology,[status(thm)],[])).
% 0.13/0.37 tff(26,plain,
% 0.13/0.37 ($false),
% 0.13/0.37 inference(unit_resolution,[status(thm)],[25, 24, 1])).
% 0.13/0.37 tff(27,plain,(disjoint(C!3, D!2) | disjoint(singleton(A!5), singleton(B!4))), inference(lemma,lemma(discharge,[]))).
% 0.13/0.37 tff(28,assumption,(~disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2))), introduced(assumption)).
% 0.13/0.37 tff(29,plain,
% 0.13/0.37 (^[A: $i, B: $i, C: $i, D: $i] : refl(((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D))) <=> ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D))))),
% 0.13/0.37 inference(bind,[status(th)],[])).
% 0.13/0.37 tff(30,plain,
% 0.13/0.37 (![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))),
% 0.13/0.37 inference(quant_intro,[status(thm)],[29])).
% 0.13/0.37 tff(31,plain,
% 0.13/0.37 (![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))),
% 0.13/0.37 inference(rewrite,[status(thm)],[])).
% 0.13/0.37 tff(32,plain,
% 0.13/0.37 (^[A: $i, B: $i, C: $i, D: $i] : trans(monotonicity(rewrite((disjoint(A, B) | disjoint(C, D)) <=> (disjoint(C, D) | disjoint(A, B))), (((disjoint(A, B) | disjoint(C, D)) => disjoint(cartesian_product2(A, C), cartesian_product2(B, D))) <=> ((disjoint(C, D) | disjoint(A, B)) => disjoint(cartesian_product2(A, C), cartesian_product2(B, D))))), rewrite(((disjoint(C, D) | disjoint(A, B)) => disjoint(cartesian_product2(A, C), cartesian_product2(B, D))) <=> ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))), (((disjoint(A, B) | disjoint(C, D)) => disjoint(cartesian_product2(A, C), cartesian_product2(B, D))) <=> ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))))),
% 0.13/0.38 inference(bind,[status(th)],[])).
% 0.13/0.38 tff(33,plain,
% 0.13/0.38 (![A: $i, B: $i, C: $i, D: $i] : ((disjoint(A, B) | disjoint(C, D)) => disjoint(cartesian_product2(A, C), cartesian_product2(B, D))) <=> ![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))),
% 0.13/0.38 inference(quant_intro,[status(thm)],[32])).
% 0.13/0.38 tff(34,axiom,(![A: $i, B: $i, C: $i, D: $i] : ((disjoint(A, B) | disjoint(C, D)) => disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t127_zfmisc_1')).
% 0.13/0.38 tff(35,plain,
% 0.13/0.38 (![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[34, 33])).
% 0.13/0.38 tff(36,plain,
% 0.13/0.38 (![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[35, 31])).
% 0.13/0.38 tff(37,plain,(
% 0.13/0.38 ![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))),
% 0.13/0.38 inference(skolemize,[status(sab)],[36])).
% 0.13/0.38 tff(38,plain,
% 0.13/0.38 (![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[37, 30])).
% 0.13/0.38 tff(39,plain,
% 0.13/0.38 (((~![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))) | ((~(disjoint(C!3, D!2) | disjoint(singleton(A!5), singleton(B!4)))) | disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2)))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))) | (~(disjoint(C!3, D!2) | disjoint(singleton(A!5), singleton(B!4)))) | disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2)))),
% 0.13/0.38 inference(rewrite,[status(thm)],[])).
% 0.13/0.38 tff(40,plain,
% 0.13/0.38 ((~![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))) | ((~(disjoint(C!3, D!2) | disjoint(singleton(A!5), singleton(B!4)))) | disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2)))),
% 0.13/0.38 inference(quant_inst,[status(thm)],[])).
% 0.13/0.38 tff(41,plain,
% 0.13/0.38 ((~![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))) | (~(disjoint(C!3, D!2) | disjoint(singleton(A!5), singleton(B!4)))) | disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[40, 39])).
% 0.13/0.38 tff(42,plain,
% 0.13/0.38 ($false),
% 0.13/0.38 inference(unit_resolution,[status(thm)],[41, 38, 28, 27])).
% 0.13/0.38 tff(43,plain,(disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2))), inference(lemma,lemma(discharge,[]))).
% 0.13/0.38 tff(44,plain,
% 0.13/0.38 ((~(~((~disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2))) | (~disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4))))))) <=> ((~disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2))) | (~disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4)))))),
% 0.13/0.38 inference(rewrite,[status(thm)],[])).
% 0.13/0.38 tff(45,plain,
% 0.13/0.38 ((disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2)) & disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4)))) <=> (~((~disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2))) | (~disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4))))))),
% 0.13/0.38 inference(rewrite,[status(thm)],[])).
% 0.13/0.38 tff(46,plain,
% 0.13/0.38 ((~(disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2)) & disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4))))) <=> (~(~((~disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2))) | (~disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4)))))))),
% 0.13/0.38 inference(monotonicity,[status(thm)],[45])).
% 0.13/0.38 tff(47,plain,
% 0.13/0.38 ((~(disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2)) & disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4))))) <=> ((~disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2))) | (~disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4)))))),
% 0.13/0.38 inference(transitivity,[status(thm)],[46, 44])).
% 0.13/0.38 tff(48,plain,
% 0.13/0.38 (~(disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2)) & disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4))))),
% 0.13/0.38 inference(or_elim,[status(thm)],[19])).
% 0.13/0.38 tff(49,plain,
% 0.13/0.38 ((~disjoint(cartesian_product2(singleton(A!5), C!3), cartesian_product2(singleton(B!4), D!2))) | (~disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4))))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[48, 47])).
% 0.13/0.38 tff(50,plain,
% 0.13/0.38 (~disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4)))),
% 0.13/0.38 inference(unit_resolution,[status(thm)],[49, 43])).
% 0.13/0.38 tff(51,plain,
% 0.13/0.38 (((~![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))) | ((~(disjoint(C!3, D!2) | disjoint(singleton(A!5), singleton(B!4)))) | disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))) | (~(disjoint(C!3, D!2) | disjoint(singleton(A!5), singleton(B!4)))) | disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4))))),
% 0.13/0.38 inference(rewrite,[status(thm)],[])).
% 0.13/0.38 tff(52,plain,
% 0.13/0.38 (((~(disjoint(singleton(A!5), singleton(B!4)) | disjoint(C!3, D!2))) | disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4)))) <=> ((~(disjoint(C!3, D!2) | disjoint(singleton(A!5), singleton(B!4)))) | disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4))))),
% 0.13/0.38 inference(rewrite,[status(thm)],[])).
% 0.13/0.38 tff(53,plain,
% 0.13/0.38 (((~![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))) | ((~(disjoint(singleton(A!5), singleton(B!4)) | disjoint(C!3, D!2))) | disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))) | ((~(disjoint(C!3, D!2) | disjoint(singleton(A!5), singleton(B!4)))) | disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4)))))),
% 0.13/0.38 inference(monotonicity,[status(thm)],[52])).
% 0.13/0.38 tff(54,plain,
% 0.13/0.38 (((~![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))) | ((~(disjoint(singleton(A!5), singleton(B!4)) | disjoint(C!3, D!2))) | disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4))))) <=> ((~![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))) | (~(disjoint(C!3, D!2) | disjoint(singleton(A!5), singleton(B!4)))) | disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4))))),
% 0.13/0.38 inference(transitivity,[status(thm)],[53, 51])).
% 0.13/0.38 tff(55,plain,
% 0.13/0.38 ((~![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))) | ((~(disjoint(singleton(A!5), singleton(B!4)) | disjoint(C!3, D!2))) | disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4))))),
% 0.13/0.38 inference(quant_inst,[status(thm)],[])).
% 0.13/0.38 tff(56,plain,
% 0.13/0.38 ((~![A: $i, B: $i, C: $i, D: $i] : ((~(disjoint(C, D) | disjoint(A, B))) | disjoint(cartesian_product2(A, C), cartesian_product2(B, D)))) | (~(disjoint(C!3, D!2) | disjoint(singleton(A!5), singleton(B!4)))) | disjoint(cartesian_product2(C!3, singleton(A!5)), cartesian_product2(D!2, singleton(B!4)))),
% 0.13/0.38 inference(modus_ponens,[status(thm)],[55, 54])).
% 0.13/0.38 tff(57,plain,
% 0.13/0.38 ($false),
% 0.13/0.38 inference(unit_resolution,[status(thm)],[56, 38, 50, 27])).
% 0.13/0.38 % SZS output end Proof
% 0.13/0.38 % E exiting
%------------------------------------------------------------------------------