%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SET008-1 : TPTP v9.0.0. Released v1.0.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n005.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:24:44 AM UTC 2025
% Result : Unsatisfiable 0.20s 0.37s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.11 % Problem : SET008-1 : TPTP v9.0.0. Released v1.0.0.
% 0.10/0.11 % Command : run_E %s %d THM
% 0.10/0.32 % Computer : n005.cluster.edu
% 0.10/0.32 % Model : x86_64 x86_64
% 0.10/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.32 % Memory : 8042.1875MB
% 0.10/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.32 % CPULimit : 300
% 0.10/0.32 % WCLimit : 300
% 0.10/0.33 % DateTime : Fri Jun 20 04:00:50 EDT 2025
% 0.10/0.33 % CPUTime :
% 0.20/0.37 % SZS status Unsatisfiable
% 0.20/0.37 % SZS output start Proof
% 0.20/0.37 tff(member_type, type, (
% 0.20/0.37 member: ( $i * $i ) > $o)).
% 0.20/0.37 tff(bDa_type, type, (
% 0.20/0.37 bDa: $i)).
% 0.20/0.37 tff(h_type, type, (
% 0.20/0.37 h: ( $i * $i * $i ) > $i)).
% 0.20/0.37 tff(aI_bDa_type, type, (
% 0.20/0.37 aI_bDa: $i)).
% 0.20/0.37 tff(a_type, type, (
% 0.20/0.37 a: $i)).
% 0.20/0.37 tff(intersection_type, type, (
% 0.20/0.37 intersection: ( $i * $i * $i ) > $o)).
% 0.20/0.37 tff(difference_type, type, (
% 0.20/0.37 difference: ( $i * $i * $i ) > $o)).
% 0.20/0.37 tff(b_type, type, (
% 0.20/0.37 b: $i)).
% 0.20/0.37 tff(1,assumption,(member(h(a, bDa, aI_bDa), aI_bDa)), introduced(assumption)).
% 0.20/0.37 tff(2,plain,
% 0.20/0.37 (^[A: $i] : refl((~member(A, aI_bDa)) <=> (~member(A, aI_bDa)))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(3,plain,
% 0.20/0.37 (![A: $i] : (~member(A, aI_bDa)) <=> ![A: $i] : (~member(A, aI_bDa))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[2])).
% 0.20/0.37 tff(4,plain,
% 0.20/0.37 (![A: $i] : (~member(A, aI_bDa)) <=> ![A: $i] : (~member(A, aI_bDa))),
% 0.20/0.37 inference(rewrite,[status(thm)],[])).
% 0.20/0.37 tff(5,axiom,(![A: $i] : (~member(A, aI_bDa))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','prove_aI_bDa_is_empty')).
% 0.20/0.37 tff(6,plain,
% 0.20/0.37 (![A: $i] : (~member(A, aI_bDa))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.20/0.37 tff(7,plain,(
% 0.20/0.37 ![A: $i] : (~member(A, aI_bDa))),
% 0.20/0.37 inference(skolemize,[status(sab)],[6])).
% 0.20/0.37 tff(8,plain,
% 0.20/0.37 (![A: $i] : (~member(A, aI_bDa))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.20/0.37 tff(9,plain,
% 0.20/0.37 ((~![A: $i] : (~member(A, aI_bDa))) | (~member(h(a, bDa, aI_bDa), aI_bDa))),
% 0.20/0.37 inference(quant_inst,[status(thm)],[])).
% 0.20/0.37 tff(10,plain,
% 0.20/0.37 ($false),
% 0.20/0.37 inference(unit_resolution,[status(thm)],[9, 8, 1])).
% 0.20/0.37 tff(11,plain,(~member(h(a, bDa, aI_bDa), aI_bDa)), inference(lemma,lemma(discharge,[]))).
% 0.20/0.37 tff(12,plain,
% 0.20/0.37 ((~intersection(a, bDa, aI_bDa)) <=> (~intersection(a, bDa, aI_bDa))),
% 0.20/0.37 inference(rewrite,[status(thm)],[])).
% 0.20/0.37 tff(13,axiom,(~intersection(a, bDa, aI_bDa)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','a_intersection_bDa')).
% 0.20/0.37 tff(14,plain,
% 0.20/0.37 (~intersection(a, bDa, aI_bDa)),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[13, 12])).
% 0.20/0.37 tff(15,plain,
% 0.20/0.37 (^[Set1: $i, Set2: $i, Intersection: $i] : refl((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)) <=> (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(16,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[15])).
% 0.20/0.37 tff(17,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.20/0.37 inference(rewrite,[status(thm)],[])).
% 0.20/0.37 tff(18,plain,
% 0.20/0.37 (^[Set1: $i, Set2: $i, Intersection: $i] : rewrite(((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set2)) <=> (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2)))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(19,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, Intersection: $i] : ((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set2)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[18])).
% 0.20/0.37 tff(20,axiom,(![Set1: $i, Set2: $i, Intersection: $i] : ((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set2))), file('/export/starexec/sandbox/benchmark/Axioms/SET001-2.ax','intersection_axiom2')).
% 0.20/0.37 tff(21,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[20, 19])).
% 0.20/0.37 tff(22,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[21, 17])).
% 0.20/0.37 tff(23,plain,(
% 0.20/0.37 ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.20/0.37 inference(skolemize,[status(sab)],[22])).
% 0.20/0.37 tff(24,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[23, 16])).
% 0.20/0.37 tff(25,plain,
% 0.20/0.37 (((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (intersection(a, bDa, aI_bDa) | member(h(a, bDa, aI_bDa), bDa) | member(h(a, bDa, aI_bDa), aI_bDa))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | intersection(a, bDa, aI_bDa) | member(h(a, bDa, aI_bDa), bDa) | member(h(a, bDa, aI_bDa), aI_bDa))),
% 0.20/0.37 inference(rewrite,[status(thm)],[])).
% 0.20/0.37 tff(26,plain,
% 0.20/0.37 ((member(h(a, bDa, aI_bDa), aI_bDa) | intersection(a, bDa, aI_bDa) | member(h(a, bDa, aI_bDa), bDa)) <=> (intersection(a, bDa, aI_bDa) | member(h(a, bDa, aI_bDa), bDa) | member(h(a, bDa, aI_bDa), aI_bDa))),
% 0.20/0.37 inference(rewrite,[status(thm)],[])).
% 0.20/0.37 tff(27,plain,
% 0.20/0.37 (((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (member(h(a, bDa, aI_bDa), aI_bDa) | intersection(a, bDa, aI_bDa) | member(h(a, bDa, aI_bDa), bDa))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (intersection(a, bDa, aI_bDa) | member(h(a, bDa, aI_bDa), bDa) | member(h(a, bDa, aI_bDa), aI_bDa)))),
% 0.20/0.37 inference(monotonicity,[status(thm)],[26])).
% 0.20/0.37 tff(28,plain,
% 0.20/0.37 (((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (member(h(a, bDa, aI_bDa), aI_bDa) | intersection(a, bDa, aI_bDa) | member(h(a, bDa, aI_bDa), bDa))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | intersection(a, bDa, aI_bDa) | member(h(a, bDa, aI_bDa), bDa) | member(h(a, bDa, aI_bDa), aI_bDa))),
% 0.20/0.37 inference(transitivity,[status(thm)],[27, 25])).
% 0.20/0.37 tff(29,plain,
% 0.20/0.37 ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | (member(h(a, bDa, aI_bDa), aI_bDa) | intersection(a, bDa, aI_bDa) | member(h(a, bDa, aI_bDa), bDa))),
% 0.20/0.37 inference(quant_inst,[status(thm)],[])).
% 0.20/0.37 tff(30,plain,
% 0.20/0.37 ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection) | member(h(Set1, Set2, Intersection), Set2))) | intersection(a, bDa, aI_bDa) | member(h(a, bDa, aI_bDa), bDa) | member(h(a, bDa, aI_bDa), aI_bDa)),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[29, 28])).
% 0.20/0.37 tff(31,plain,
% 0.20/0.37 (member(h(a, bDa, aI_bDa), bDa) | member(h(a, bDa, aI_bDa), aI_bDa)),
% 0.20/0.37 inference(unit_resolution,[status(thm)],[30, 24, 14])).
% 0.20/0.37 tff(32,plain,
% 0.20/0.37 (member(h(a, bDa, aI_bDa), bDa)),
% 0.20/0.37 inference(unit_resolution,[status(thm)],[31, 11])).
% 0.20/0.37 tff(33,plain,
% 0.20/0.37 (^[Set1: $i, Set2: $i, Intersection: $i] : refl((member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) <=> (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(34,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[33])).
% 0.20/0.37 tff(35,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.20/0.37 inference(rewrite,[status(thm)],[])).
% 0.20/0.37 tff(36,plain,
% 0.20/0.37 (^[Set1: $i, Set2: $i, Intersection: $i] : rewrite(((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set1)) <=> (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(37,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, Intersection: $i] : ((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set1)) <=> ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[36])).
% 0.20/0.37 tff(38,axiom,(![Set1: $i, Set2: $i, Intersection: $i] : ((member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection)) | member(h(Set1, Set2, Intersection), Set1))), file('/export/starexec/sandbox/benchmark/Axioms/SET001-2.ax','intersection_axiom1')).
% 0.20/0.37 tff(39,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[38, 37])).
% 0.20/0.37 tff(40,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[39, 35])).
% 0.20/0.37 tff(41,plain,(
% 0.20/0.37 ![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.20/0.37 inference(skolemize,[status(sab)],[40])).
% 0.20/0.37 tff(42,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[41, 34])).
% 0.20/0.37 tff(43,plain,
% 0.20/0.37 (((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))) | (member(h(a, bDa, aI_bDa), a) | member(h(a, bDa, aI_bDa), aI_bDa) | intersection(a, bDa, aI_bDa))) <=> ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))) | member(h(a, bDa, aI_bDa), a) | member(h(a, bDa, aI_bDa), aI_bDa) | intersection(a, bDa, aI_bDa))),
% 0.20/0.37 inference(rewrite,[status(thm)],[])).
% 0.20/0.37 tff(44,plain,
% 0.20/0.37 ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))) | (member(h(a, bDa, aI_bDa), a) | member(h(a, bDa, aI_bDa), aI_bDa) | intersection(a, bDa, aI_bDa))),
% 0.20/0.37 inference(quant_inst,[status(thm)],[])).
% 0.20/0.37 tff(45,plain,
% 0.20/0.37 ((~![Set1: $i, Set2: $i, Intersection: $i] : (member(h(Set1, Set2, Intersection), Set1) | member(h(Set1, Set2, Intersection), Intersection) | intersection(Set1, Set2, Intersection))) | member(h(a, bDa, aI_bDa), a) | member(h(a, bDa, aI_bDa), aI_bDa) | intersection(a, bDa, aI_bDa)),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[44, 43])).
% 0.20/0.37 tff(46,plain,
% 0.20/0.37 (member(h(a, bDa, aI_bDa), a) | member(h(a, bDa, aI_bDa), aI_bDa)),
% 0.20/0.37 inference(unit_resolution,[status(thm)],[45, 42, 14])).
% 0.20/0.37 tff(47,plain,
% 0.20/0.37 (member(h(a, bDa, aI_bDa), a)),
% 0.20/0.37 inference(unit_resolution,[status(thm)],[46, 11])).
% 0.20/0.37 tff(48,plain,
% 0.20/0.37 (difference(b, a, bDa) <=> difference(b, a, bDa)),
% 0.20/0.37 inference(rewrite,[status(thm)],[])).
% 0.20/0.37 tff(49,axiom,(difference(b, a, bDa)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','b_minus_a')).
% 0.20/0.37 tff(50,plain,
% 0.20/0.37 (difference(b, a, bDa)),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[49, 48])).
% 0.20/0.37 tff(51,plain,
% 0.20/0.37 (^[Set1: $i, Set2: $i, A_set: $i, Element: $i] : refl(((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1))) <=> ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1))))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(52,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1))) <=> ![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[51])).
% 0.20/0.37 tff(53,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1))) <=> ![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.20/0.37 inference(rewrite,[status(thm)],[])).
% 0.20/0.37 tff(54,plain,
% 0.20/0.37 (^[Set1: $i, Set2: $i, A_set: $i, Element: $i] : trans(monotonicity(rewrite(((~member(Element, Set1)) | (~member(Element, Set2))) <=> ((~member(Element, Set2)) | (~member(Element, Set1)))), ((((~member(Element, Set1)) | (~member(Element, Set2))) | (~difference(A_set, Set1, Set2))) <=> (((~member(Element, Set2)) | (~member(Element, Set1))) | (~difference(A_set, Set1, Set2))))), rewrite((((~member(Element, Set2)) | (~member(Element, Set1))) | (~difference(A_set, Set1, Set2))) <=> ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))), ((((~member(Element, Set1)) | (~member(Element, Set2))) | (~difference(A_set, Set1, Set2))) <=> ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))))),
% 0.20/0.37 inference(bind,[status(th)],[])).
% 0.20/0.37 tff(55,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, A_set: $i, Element: $i] : (((~member(Element, Set1)) | (~member(Element, Set2))) | (~difference(A_set, Set1, Set2))) <=> ![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.20/0.37 inference(quant_intro,[status(thm)],[54])).
% 0.20/0.37 tff(56,axiom,(![Set1: $i, Set2: $i, A_set: $i, Element: $i] : (((~member(Element, Set1)) | (~member(Element, Set2))) | (~difference(A_set, Set1, Set2)))), file('/export/starexec/sandbox/benchmark/Axioms/SET001-3.ax','not_member_of_difference')).
% 0.20/0.37 tff(57,plain,
% 0.20/0.37 (![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.20/0.37 inference(modus_ponens,[status(thm)],[56, 55])).
% 0.20/0.38 tff(58,plain,
% 0.20/0.38 (![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[57, 53])).
% 0.20/0.38 tff(59,plain,(
% 0.20/0.38 ![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.20/0.38 inference(skolemize,[status(sab)],[58])).
% 0.20/0.38 tff(60,plain,
% 0.20/0.38 (![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[59, 52])).
% 0.20/0.38 tff(61,plain,
% 0.20/0.38 (((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | ((~member(h(a, bDa, aI_bDa), a)) | (~member(h(a, bDa, aI_bDa), bDa)) | (~difference(b, a, bDa)))) <=> ((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | (~member(h(a, bDa, aI_bDa), a)) | (~member(h(a, bDa, aI_bDa), bDa)) | (~difference(b, a, bDa)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(62,plain,
% 0.20/0.38 (((~difference(b, a, bDa)) | (~member(h(a, bDa, aI_bDa), bDa)) | (~member(h(a, bDa, aI_bDa), a))) <=> ((~member(h(a, bDa, aI_bDa), a)) | (~member(h(a, bDa, aI_bDa), bDa)) | (~difference(b, a, bDa)))),
% 0.20/0.38 inference(rewrite,[status(thm)],[])).
% 0.20/0.38 tff(63,plain,
% 0.20/0.38 (((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | ((~difference(b, a, bDa)) | (~member(h(a, bDa, aI_bDa), bDa)) | (~member(h(a, bDa, aI_bDa), a)))) <=> ((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | ((~member(h(a, bDa, aI_bDa), a)) | (~member(h(a, bDa, aI_bDa), bDa)) | (~difference(b, a, bDa))))),
% 0.20/0.38 inference(monotonicity,[status(thm)],[62])).
% 0.20/0.38 tff(64,plain,
% 0.20/0.38 (((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | ((~difference(b, a, bDa)) | (~member(h(a, bDa, aI_bDa), bDa)) | (~member(h(a, bDa, aI_bDa), a)))) <=> ((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | (~member(h(a, bDa, aI_bDa), a)) | (~member(h(a, bDa, aI_bDa), bDa)) | (~difference(b, a, bDa)))),
% 0.20/0.38 inference(transitivity,[status(thm)],[63, 61])).
% 0.20/0.38 tff(65,plain,
% 0.20/0.38 ((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | ((~difference(b, a, bDa)) | (~member(h(a, bDa, aI_bDa), bDa)) | (~member(h(a, bDa, aI_bDa), a)))),
% 0.20/0.38 inference(quant_inst,[status(thm)],[])).
% 0.20/0.38 tff(66,plain,
% 0.20/0.38 ((~![Set1: $i, Set2: $i, A_set: $i, Element: $i] : ((~difference(A_set, Set1, Set2)) | (~member(Element, Set2)) | (~member(Element, Set1)))) | (~member(h(a, bDa, aI_bDa), a)) | (~member(h(a, bDa, aI_bDa), bDa)) | (~difference(b, a, bDa))),
% 0.20/0.38 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.20/0.38 tff(67,plain,
% 0.20/0.38 ($false),
% 0.20/0.38 inference(unit_resolution,[status(thm)],[66, 60, 50, 47, 32])).
% 0.20/0.38 % SZS output end Proof
% 0.20/0.38 % E exiting
%------------------------------------------------------------------------------