%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU062+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n003.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 05:26:51 AM UTC 2025
% Result : Theorem 0.12s 0.39s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : SEU062+1 : TPTP v9.0.0. Released v3.2.0.
% 0.07/0.13 % Command : run_E %s %d THM
% 0.12/0.34 % Computer : n003.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Jun 20 06:02:09 EDT 2025
% 0.12/0.34 % CPUTime :
% 0.12/0.39 % SZS status Theorem
% 0.12/0.39 % SZS output start Proof
% 0.12/0.39 tff(in_type, type, (
% 0.12/0.39 in: ( $i * $i ) > $o)).
% 0.12/0.39 tff(relation_rng_type, type, (
% 0.12/0.39 relation_rng: $i > $i)).
% 0.12/0.39 tff(tptp_fun_B_12_type, type, (
% 0.12/0.39 tptp_fun_B_12: $i)).
% 0.12/0.39 tff(tptp_fun_C_0_type, type, (
% 0.12/0.39 tptp_fun_C_0: ( $i * $i ) > $i)).
% 0.12/0.39 tff(tptp_fun_A_13_type, type, (
% 0.12/0.39 tptp_fun_A_13: $i)).
% 0.12/0.39 tff(empty_set_type, type, (
% 0.12/0.39 empty_set: $i)).
% 0.12/0.39 tff(relation_inverse_image_type, type, (
% 0.12/0.39 relation_inverse_image: ( $i * $i ) > $i)).
% 0.12/0.39 tff(singleton_type, type, (
% 0.12/0.39 singleton: $i > $i)).
% 0.12/0.39 tff(relation_type, type, (
% 0.12/0.39 relation: $i > $o)).
% 0.12/0.39 tff(subset_type, type, (
% 0.12/0.39 subset: ( $i * $i ) > $o)).
% 0.12/0.39 tff(1,plain,
% 0.12/0.39 (((~subset(A!13, relation_rng(B!12))) & relation(B!12) & ![C: $i] : (~(in(C, A!13) & (relation_inverse_image(B!12, singleton(C)) = empty_set)))) <=> ((~subset(A!13, relation_rng(B!12))) & relation(B!12) & ![C: $i] : (~(in(C, A!13) & (relation_inverse_image(B!12, singleton(C)) = empty_set))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(2,plain,
% 0.12/0.39 ((~![A: $i, B: $i] : (subset(A, relation_rng(B)) | (~relation(B)) | (~![C: $i] : (~(in(C, A) & (relation_inverse_image(B, singleton(C)) = empty_set)))))) <=> (~![A: $i, B: $i] : (subset(A, relation_rng(B)) | (~relation(B)) | (~![C: $i] : (~(in(C, A) & (relation_inverse_image(B, singleton(C)) = empty_set))))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(3,plain,
% 0.12/0.39 ((~![A: $i, B: $i] : (relation(B) => (![C: $i] : (~(in(C, A) & (relation_inverse_image(B, singleton(C)) = empty_set))) => subset(A, relation_rng(B))))) <=> (~![A: $i, B: $i] : (subset(A, relation_rng(B)) | (~relation(B)) | (~![C: $i] : (~(in(C, A) & (relation_inverse_image(B, singleton(C)) = empty_set))))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(4,axiom,(~![A: $i, B: $i] : (relation(B) => (![C: $i] : (~(in(C, A) & (relation_inverse_image(B, singleton(C)) = empty_set))) => subset(A, relation_rng(B))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t143_funct_1')).
% 0.12/0.39 tff(5,plain,
% 0.12/0.39 (~![A: $i, B: $i] : (subset(A, relation_rng(B)) | (~relation(B)) | (~![C: $i] : (~(in(C, A) & (relation_inverse_image(B, singleton(C)) = empty_set)))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[4, 3])).
% 0.12/0.39 tff(6,plain,
% 0.12/0.39 (~![A: $i, B: $i] : (subset(A, relation_rng(B)) | (~relation(B)) | (~![C: $i] : (~(in(C, A) & (relation_inverse_image(B, singleton(C)) = empty_set)))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[5, 2])).
% 0.12/0.39 tff(7,plain,
% 0.12/0.39 (~![A: $i, B: $i] : (subset(A, relation_rng(B)) | (~relation(B)) | (~![C: $i] : (~(in(C, A) & (relation_inverse_image(B, singleton(C)) = empty_set)))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[6, 2])).
% 0.12/0.39 tff(8,plain,
% 0.12/0.39 (~![A: $i, B: $i] : (subset(A, relation_rng(B)) | (~relation(B)) | (~![C: $i] : (~(in(C, A) & (relation_inverse_image(B, singleton(C)) = empty_set)))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[7, 2])).
% 0.12/0.39 tff(9,plain,
% 0.12/0.39 ((~subset(A!13, relation_rng(B!12))) & relation(B!12) & ![C: $i] : (~(in(C, A!13) & (relation_inverse_image(B!12, singleton(C)) = empty_set)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[8, 1])).
% 0.12/0.39 tff(10,plain,
% 0.12/0.39 (relation(B!12)),
% 0.12/0.39 inference(and_elim,[status(thm)],[9])).
% 0.12/0.39 tff(11,plain,
% 0.12/0.39 (^[A: $i, B: $i] : refl(((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B))))) <=> ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(12,plain,
% 0.12/0.39 (![A: $i, B: $i] : ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B))))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[11])).
% 0.12/0.39 tff(13,plain,
% 0.12/0.39 (![A: $i, B: $i] : ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B))))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(14,plain,
% 0.12/0.39 (^[A: $i, B: $i] : trans(monotonicity(trans(rewrite((in(A, relation_rng(B)) <=> (~(relation_inverse_image(B, singleton(A)) = empty_set))) <=> (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B))))), rewrite((~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))) <=> (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B))))), ((in(A, relation_rng(B)) <=> (~(relation_inverse_image(B, singleton(A)) = empty_set))) <=> (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))), ((relation(B) => (in(A, relation_rng(B)) <=> (~(relation_inverse_image(B, singleton(A)) = empty_set)))) <=> (relation(B) => (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B))))))), rewrite((relation(B) => (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B))))) <=> ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))), ((relation(B) => (in(A, relation_rng(B)) <=> (~(relation_inverse_image(B, singleton(A)) = empty_set)))) <=> ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(15,plain,
% 0.12/0.39 (![A: $i, B: $i] : (relation(B) => (in(A, relation_rng(B)) <=> (~(relation_inverse_image(B, singleton(A)) = empty_set)))) <=> ![A: $i, B: $i] : ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[14])).
% 0.12/0.39 tff(16,axiom,(![A: $i, B: $i] : (relation(B) => (in(A, relation_rng(B)) <=> (~(relation_inverse_image(B, singleton(A)) = empty_set))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t142_funct_1')).
% 0.12/0.39 tff(17,plain,
% 0.12/0.39 (![A: $i, B: $i] : ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.12/0.39 tff(18,plain,
% 0.12/0.39 (![A: $i, B: $i] : ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[17, 13])).
% 0.12/0.39 tff(19,plain,(
% 0.12/0.39 ![A: $i, B: $i] : ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))),
% 0.12/0.39 inference(skolemize,[status(sab)],[18])).
% 0.12/0.39 tff(20,plain,
% 0.12/0.39 (![A: $i, B: $i] : ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[19, 12])).
% 0.12/0.39 tff(21,plain,
% 0.12/0.39 (((~![A: $i, B: $i] : ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))) | ((~relation(B!12)) | (~((relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set) <=> in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12)))))) <=> ((~![A: $i, B: $i] : ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))) | (~relation(B!12)) | (~((relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set) <=> in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12)))))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(22,plain,
% 0.12/0.39 ((~![A: $i, B: $i] : ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))) | ((~relation(B!12)) | (~((relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set) <=> in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12)))))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(23,plain,
% 0.12/0.39 ((~![A: $i, B: $i] : ((~relation(B)) | (~((relation_inverse_image(B, singleton(A)) = empty_set) <=> in(A, relation_rng(B)))))) | (~relation(B!12)) | (~((relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set) <=> in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.12/0.39 tff(24,plain,
% 0.12/0.39 (~((relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set) <=> in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12)))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[23, 20, 10])).
% 0.12/0.39 tff(25,plain,
% 0.12/0.39 (^[A: $i, B: $i] : refl((~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(26,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[25])).
% 0.12/0.39 tff(27,plain,
% 0.12/0.39 (^[A: $i, B: $i] : rewrite((~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(28,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[27])).
% 0.12/0.39 tff(29,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))),
% 0.12/0.39 inference(transitivity,[status(thm)],[28, 26])).
% 0.12/0.39 tff(30,plain,
% 0.12/0.39 (^[A: $i, B: $i] : trans(monotonicity(rewrite(((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) <=> ((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))), rewrite((subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))) <=> (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))), ((((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))) <=> (((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))))), rewrite((((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))), ((((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))) <=> (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(31,plain,
% 0.12/0.39 (![A: $i, B: $i] : (((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B))))) <=> ![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[30])).
% 0.12/0.39 tff(32,plain,
% 0.12/0.39 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 0.12/0.39 inference(rewrite,[status(thm)],[])).
% 0.12/0.39 tff(33,plain,
% 0.12/0.39 (^[A: $i, B: $i] : rewrite((subset(A, B) <=> ![C: $i] : (in(C, A) => in(C, B))) <=> (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B))))),
% 0.12/0.39 inference(bind,[status(th)],[])).
% 0.12/0.39 tff(34,plain,
% 0.12/0.39 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : (in(C, A) => in(C, B))) <=> ![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 0.12/0.39 inference(quant_intro,[status(thm)],[33])).
% 0.12/0.39 tff(35,axiom,(![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : (in(C, A) => in(C, B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d3_tarski')).
% 0.12/0.39 tff(36,plain,
% 0.12/0.39 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.12/0.39 tff(37,plain,
% 0.12/0.39 (![A: $i, B: $i] : (subset(A, B) <=> ![C: $i] : ((~in(C, A)) | in(C, B)))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.12/0.39 tff(38,plain,(
% 0.12/0.39 ![A: $i, B: $i] : (((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B))) & (subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))),
% 0.12/0.39 inference(skolemize,[status(sab)],[37])).
% 0.12/0.39 tff(39,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[38, 31])).
% 0.12/0.39 tff(40,plain,
% 0.12/0.39 (![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))),
% 0.12/0.39 inference(modus_ponens,[status(thm)],[39, 29])).
% 0.12/0.39 tff(41,plain,
% 0.12/0.39 ((~![A: $i, B: $i] : (~((~((~subset(A, B)) | ![C: $i] : ((~in(C, A)) | in(C, B)))) | (~(subset(A, B) | (~((~in(tptp_fun_C_0(B, A), A)) | in(tptp_fun_C_0(B, A), B)))))))) | (~((~((~subset(A!13, relation_rng(B!12))) | ![C: $i] : ((~in(C, A!13)) | in(C, relation_rng(B!12))))) | (~(subset(A!13, relation_rng(B!12)) | (~((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12))))))))),
% 0.12/0.39 inference(quant_inst,[status(thm)],[])).
% 0.12/0.39 tff(42,plain,
% 0.12/0.39 (~((~((~subset(A!13, relation_rng(B!12))) | ![C: $i] : ((~in(C, A!13)) | in(C, relation_rng(B!12))))) | (~(subset(A!13, relation_rng(B!12)) | (~((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12)))))))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[41, 40])).
% 0.12/0.39 tff(43,plain,
% 0.12/0.39 (((~((~subset(A!13, relation_rng(B!12))) | ![C: $i] : ((~in(C, A!13)) | in(C, relation_rng(B!12))))) | (~(subset(A!13, relation_rng(B!12)) | (~((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12))))))) | (subset(A!13, relation_rng(B!12)) | (~((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12)))))),
% 0.12/0.39 inference(tautology,[status(thm)],[])).
% 0.12/0.39 tff(44,plain,
% 0.12/0.39 (subset(A!13, relation_rng(B!12)) | (~((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12))))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[43, 42])).
% 0.12/0.39 tff(45,plain,
% 0.12/0.39 (~subset(A!13, relation_rng(B!12))),
% 0.12/0.39 inference(and_elim,[status(thm)],[9])).
% 0.12/0.39 tff(46,plain,
% 0.12/0.39 ((~(subset(A!13, relation_rng(B!12)) | (~((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12)))))) | subset(A!13, relation_rng(B!12)) | (~((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12))))),
% 0.12/0.39 inference(tautology,[status(thm)],[])).
% 0.12/0.39 tff(47,plain,
% 0.12/0.39 ((~(subset(A!13, relation_rng(B!12)) | (~((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12)))))) | (~((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12))))),
% 0.12/0.39 inference(unit_resolution,[status(thm)],[46, 45])).
% 0.12/0.39 tff(48,plain,
% 0.12/0.39 (~((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12)))),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[47, 44])).
% 0.12/0.40 tff(49,plain,
% 0.12/0.40 (((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12))) | in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(50,plain,
% 0.12/0.40 (in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[49, 48])).
% 0.12/0.40 tff(51,plain,
% 0.12/0.40 (^[C: $i] : refl(((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set))) <=> ((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(52,plain,
% 0.12/0.40 (![C: $i] : ((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set))) <=> ![C: $i] : ((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[51])).
% 0.12/0.40 tff(53,plain,
% 0.12/0.40 (^[C: $i] : trans(monotonicity(rewrite((in(C, A!13) & (relation_inverse_image(B!12, singleton(C)) = empty_set)) <=> (~((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set))))), ((~(in(C, A!13) & (relation_inverse_image(B!12, singleton(C)) = empty_set))) <=> (~(~((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set))))))), rewrite((~(~((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set))))) <=> ((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set)))), ((~(in(C, A!13) & (relation_inverse_image(B!12, singleton(C)) = empty_set))) <=> ((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set)))))),
% 0.12/0.40 inference(bind,[status(th)],[])).
% 0.12/0.40 tff(54,plain,
% 0.12/0.40 (![C: $i] : (~(in(C, A!13) & (relation_inverse_image(B!12, singleton(C)) = empty_set))) <=> ![C: $i] : ((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set)))),
% 0.12/0.40 inference(quant_intro,[status(thm)],[53])).
% 0.12/0.40 tff(55,plain,
% 0.12/0.40 (![C: $i] : (~(in(C, A!13) & (relation_inverse_image(B!12, singleton(C)) = empty_set)))),
% 0.12/0.40 inference(and_elim,[status(thm)],[9])).
% 0.12/0.40 tff(56,plain,
% 0.12/0.40 (![C: $i] : ((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[55, 54])).
% 0.12/0.40 tff(57,plain,
% 0.12/0.40 (![C: $i] : ((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set)))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[56, 52])).
% 0.12/0.40 tff(58,plain,
% 0.12/0.40 (((~![C: $i] : ((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set)))) | ((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | (~(relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set)))) <=> ((~![C: $i] : ((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set)))) | (~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | (~(relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set)))),
% 0.12/0.40 inference(rewrite,[status(thm)],[])).
% 0.12/0.40 tff(59,plain,
% 0.12/0.40 ((~![C: $i] : ((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set)))) | ((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | (~(relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set)))),
% 0.12/0.40 inference(quant_inst,[status(thm)],[])).
% 0.12/0.40 tff(60,plain,
% 0.12/0.40 ((~![C: $i] : ((~in(C, A!13)) | (~(relation_inverse_image(B!12, singleton(C)) = empty_set)))) | (~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | (~(relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set))),
% 0.12/0.40 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.12/0.40 tff(61,plain,
% 0.12/0.40 (~(relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set)),
% 0.12/0.40 inference(unit_resolution,[status(thm)],[60, 57, 50])).
% 0.12/0.40 tff(62,plain,
% 0.12/0.40 (((~in(tptp_fun_C_0(relation_rng(B!12), A!13), A!13)) | in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12))) | (~in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12)))),
% 0.12/0.40 inference(tautology,[status(thm)],[])).
% 0.12/0.40 tff(63,plain,
% 0.12/0.40 (~in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[62, 48])).
% 0.21/0.40 tff(64,plain,
% 0.21/0.40 (((relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set) <=> in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12))) | (relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set) | in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12))),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(65,plain,
% 0.21/0.40 (((relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set) <=> in(tptp_fun_C_0(relation_rng(B!12), A!13), relation_rng(B!12))) | (relation_inverse_image(B!12, singleton(tptp_fun_C_0(relation_rng(B!12), A!13))) = empty_set)),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[64, 63])).
% 0.21/0.40 tff(66,plain,
% 0.21/0.40 ($false),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[65, 61, 24])).
% 0.21/0.40 % SZS output end Proof
% 0.21/0.40 % E exiting
%------------------------------------------------------------------------------