TSTP Solution File: NUM537+2 by SuperZenon---0.0.1
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- Process Solution
%------------------------------------------------------------------------------
% File : SuperZenon---0.0.1
% Problem : NUM537+2 : TPTP v8.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% Computer : n025.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 : 600s
% DateTime : Mon Jul 18 14:43:24 EDT 2022
% Result : Theorem 1.40s 1.63s
% Output : Proof 1.40s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.11 % Problem : NUM537+2 : TPTP v8.1.0. Released v4.0.0.
% 0.07/0.12 % Command : run_super_zenon -p0 -itptp -om -max-time %d %s
% 0.12/0.33 % Computer : n025.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 600
% 0.12/0.33 % DateTime : Wed Jul 6 00:30:27 EDT 2022
% 0.12/0.33 % CPUTime :
% 1.40/1.63 % SZS status Theorem
% 1.40/1.63 (* PROOF-FOUND *)
% 1.40/1.63 (* BEGIN-PROOF *)
% 1.40/1.63 % SZS output start Proof
% 1.40/1.63 1. (aSet0 (xS)) (-. (aSet0 (xS))) ### Axiom
% 1.40/1.63 2. (aSet0 (sdtpldt0 (xS) (xx))) (-. (aSet0 (sdtpldt0 (xS) (xx)))) ### Axiom
% 1.40/1.63 3. (aSet0 (xS)) (-. (aSet0 (xS))) ### Axiom
% 1.40/1.63 4. (aElementOf0 T_0 (xS)) (-. (aElementOf0 T_0 (xS))) ### Axiom
% 1.40/1.63 5. (-. (aElement0 T_0)) (aElement0 T_0) ### Axiom
% 1.40/1.63 6. ((aElementOf0 T_0 (xS)) => (aElement0 T_0)) (-. (aElement0 T_0)) (aElementOf0 T_0 (xS)) ### Imply 4 5
% 1.40/1.63 7. (All W1, ((aElementOf0 W1 (xS)) => (aElement0 W1))) (aElementOf0 T_0 (xS)) (-. (aElement0 T_0)) ### All 6
% 1.40/1.63 8. (aElementOf0 T_0 (xS)) (-. (aElementOf0 T_0 (xS))) ### Axiom
% 1.40/1.63 9. (-. ((aElementOf0 T_0 (xS)) \/ (T_0 = (xx)))) (aElementOf0 T_0 (xS)) ### NotOr 8
% 1.40/1.63 10. (-. ((aElement0 T_0) /\ ((aElementOf0 T_0 (xS)) \/ (T_0 = (xx))))) (aElementOf0 T_0 (xS)) (All W1, ((aElementOf0 W1 (xS)) => (aElement0 W1))) ### NotAnd 7 9
% 1.40/1.63 11. (-. (aElementOf0 T_0 (sdtpldt0 (xS) (xx)))) (aElementOf0 T_0 (sdtpldt0 (xS) (xx))) ### Axiom
% 1.40/1.63 12. ((aElementOf0 T_0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 T_0) /\ ((aElementOf0 T_0 (xS)) \/ (T_0 = (xx))))) (-. (aElementOf0 T_0 (sdtpldt0 (xS) (xx)))) (All W1, ((aElementOf0 W1 (xS)) => (aElement0 W1))) (aElementOf0 T_0 (xS)) ### Equiv 10 11
% 1.40/1.63 13. (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (aElementOf0 T_0 (xS)) (All W1, ((aElementOf0 W1 (xS)) => (aElement0 W1))) (-. (aElementOf0 T_0 (sdtpldt0 (xS) (xx)))) ### All 12
% 1.40/1.63 14. (-. ((aElementOf0 T_0 (xS)) => (aElementOf0 T_0 (sdtpldt0 (xS) (xx))))) (All W1, ((aElementOf0 W1 (xS)) => (aElement0 W1))) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) ### NotImply 13
% 1.40/1.63 15. (-. (All W2, ((aElementOf0 W2 (xS)) => (aElementOf0 W2 (sdtpldt0 (xS) (xx)))))) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (All W1, ((aElementOf0 W1 (xS)) => (aElement0 W1))) ### NotAllEx 14
% 1.40/1.63 16. (-. ((aSet0 (xS)) /\ (All W2, ((aElementOf0 W2 (xS)) => (aElementOf0 W2 (sdtpldt0 (xS) (xx))))))) (All W1, ((aElementOf0 W1 (xS)) => (aElement0 W1))) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (aSet0 (xS)) ### NotAnd 3 15
% 1.40/1.63 17. (aElementOf0 T_1 (xS)) (-. (aElementOf0 T_1 (xS))) ### Axiom
% 1.40/1.63 18. (-. (aElementOf0 T_1 (sdtpldt0 (xS) (xx)))) (aElementOf0 T_1 (sdtpldt0 (xS) (xx))) ### Axiom
% 1.40/1.63 19. ((aElementOf0 T_1 (xS)) => (aElementOf0 T_1 (sdtpldt0 (xS) (xx)))) (-. (aElementOf0 T_1 (sdtpldt0 (xS) (xx)))) (aElementOf0 T_1 (xS)) ### Imply 17 18
% 1.40/1.63 20. (All W2, ((aElementOf0 W2 (xS)) => (aElementOf0 W2 (sdtpldt0 (xS) (xx))))) (aElementOf0 T_1 (xS)) (-. (aElementOf0 T_1 (sdtpldt0 (xS) (xx)))) ### All 19
% 1.40/1.63 21. ((aSet0 (xS)) /\ (All W2, ((aElementOf0 W2 (xS)) => (aElementOf0 W2 (sdtpldt0 (xS) (xx)))))) (-. (aElementOf0 T_1 (sdtpldt0 (xS) (xx)))) (aElementOf0 T_1 (xS)) ### And 20
% 1.40/1.63 22. ((aSubsetOf0 (xS) (sdtpldt0 (xS) (xx))) <=> ((aSet0 (xS)) /\ (All W2, ((aElementOf0 W2 (xS)) => (aElementOf0 W2 (sdtpldt0 (xS) (xx))))))) (aElementOf0 T_1 (xS)) (-. (aElementOf0 T_1 (sdtpldt0 (xS) (xx)))) (aSet0 (xS)) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (All W1, ((aElementOf0 W1 (xS)) => (aElement0 W1))) ### Equiv 16 21
% 1.40/1.63 23. (All W1, ((aSubsetOf0 W1 (sdtpldt0 (xS) (xx))) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 (sdtpldt0 (xS) (xx)))))))) (All W1, ((aElementOf0 W1 (xS)) => (aElement0 W1))) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (aSet0 (xS)) (-. (aElementOf0 T_1 (sdtpldt0 (xS) (xx)))) (aElementOf0 T_1 (xS)) ### All 22
% 1.40/1.63 24. ((aSet0 (sdtpldt0 (xS) (xx))) => (All W1, ((aSubsetOf0 W1 (sdtpldt0 (xS) (xx))) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 (sdtpldt0 (xS) (xx))))))))) (aElementOf0 T_1 (xS)) (-. (aElementOf0 T_1 (sdtpldt0 (xS) (xx)))) (aSet0 (xS)) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (All W1, ((aElementOf0 W1 (xS)) => (aElement0 W1))) (aSet0 (sdtpldt0 (xS) (xx))) ### Imply 2 23
% 1.40/1.63 25. (All W0, ((aSet0 W0) => (All W1, ((aSubsetOf0 W1 W0) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 W0)))))))) (aSet0 (sdtpldt0 (xS) (xx))) (All W1, ((aElementOf0 W1 (xS)) => (aElement0 W1))) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (aSet0 (xS)) (-. (aElementOf0 T_1 (sdtpldt0 (xS) (xx)))) (aElementOf0 T_1 (xS)) ### All 24
% 1.40/1.63 26. ((aSet0 (xS)) => (All W1, ((aElementOf0 W1 (xS)) => (aElement0 W1)))) (aElementOf0 T_1 (xS)) (-. (aElementOf0 T_1 (sdtpldt0 (xS) (xx)))) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (aSet0 (sdtpldt0 (xS) (xx))) (All W0, ((aSet0 W0) => (All W1, ((aSubsetOf0 W1 W0) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 W0)))))))) (aSet0 (xS)) ### Imply 1 25
% 1.40/1.63 27. (All W0, ((aSet0 W0) => (All W1, ((aElementOf0 W1 W0) => (aElement0 W1))))) (aSet0 (xS)) (All W0, ((aSet0 W0) => (All W1, ((aSubsetOf0 W1 W0) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 W0)))))))) (aSet0 (sdtpldt0 (xS) (xx))) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (-. (aElementOf0 T_1 (sdtpldt0 (xS) (xx)))) (aElementOf0 T_1 (xS)) ### All 26
% 1.40/1.63 28. (-. (aElement0 T_1)) (aElement0 T_1) ### Axiom
% 1.40/1.63 29. ((aElement0 T_1) /\ ((aElementOf0 T_1 (xS)) \/ (T_1 = (xx)))) (-. (aElement0 T_1)) ### And 28
% 1.40/1.63 30. ((aElementOf0 T_1 (sdtpldt0 (xS) (xx))) <=> ((aElement0 T_1) /\ ((aElementOf0 T_1 (xS)) \/ (T_1 = (xx))))) (-. (aElement0 T_1)) (aElementOf0 T_1 (xS)) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (aSet0 (sdtpldt0 (xS) (xx))) (All W0, ((aSet0 W0) => (All W1, ((aSubsetOf0 W1 W0) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 W0)))))))) (aSet0 (xS)) (All W0, ((aSet0 W0) => (All W1, ((aElementOf0 W1 W0) => (aElement0 W1))))) ### Equiv 27 29
% 1.40/1.63 31. (All W0, ((aSet0 W0) => (All W1, ((aElementOf0 W1 W0) => (aElement0 W1))))) (aSet0 (xS)) (All W0, ((aSet0 W0) => (All W1, ((aSubsetOf0 W1 W0) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 W0)))))))) (aSet0 (sdtpldt0 (xS) (xx))) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (aElementOf0 T_1 (xS)) (-. (aElement0 T_1)) ### All 30
% 1.40/1.63 32. (T_1 = (xx)) (T_1 != (xx)) ### Axiom
% 1.40/1.63 33. ((xS) != (xS)) ### NotEqual
% 1.40/1.63 34. (-. (aElementOf0 (xx) (xS))) (aElementOf0 T_1 (xS)) (T_1 = (xx)) ### P-NotP 32 33
% 1.40/1.63 35. (-. ((aElement0 T_1) /\ ((aElementOf0 T_1 (sdtpldt0 (xS) (xx))) /\ (T_1 != (xx))))) (-. (aElementOf0 (xx) (xS))) (aElementOf0 T_1 (xS)) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (aSet0 (sdtpldt0 (xS) (xx))) (All W0, ((aSet0 W0) => (All W1, ((aSubsetOf0 W1 W0) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 W0)))))))) (aSet0 (xS)) (All W0, ((aSet0 W0) => (All W1, ((aElementOf0 W1 W0) => (aElement0 W1))))) ### DisjTree 31 27 34
% 1.40/1.63 36. (-. (aElementOf0 T_1 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx)))) (aElementOf0 T_1 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) ### Axiom
% 1.40/1.63 37. ((aElementOf0 T_1 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 T_1) /\ ((aElementOf0 T_1 (sdtpldt0 (xS) (xx))) /\ (T_1 != (xx))))) (-. (aElementOf0 T_1 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx)))) (All W0, ((aSet0 W0) => (All W1, ((aElementOf0 W1 W0) => (aElement0 W1))))) (aSet0 (xS)) (All W0, ((aSet0 W0) => (All W1, ((aSubsetOf0 W1 W0) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 W0)))))))) (aSet0 (sdtpldt0 (xS) (xx))) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (aElementOf0 T_1 (xS)) (-. (aElementOf0 (xx) (xS))) ### Equiv 35 36
% 1.40/1.63 38. (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx)))))) (-. (aElementOf0 (xx) (xS))) (aElementOf0 T_1 (xS)) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (aSet0 (sdtpldt0 (xS) (xx))) (All W0, ((aSet0 W0) => (All W1, ((aSubsetOf0 W1 W0) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 W0)))))))) (aSet0 (xS)) (All W0, ((aSet0 W0) => (All W1, ((aElementOf0 W1 W0) => (aElement0 W1))))) (-. (aElementOf0 T_1 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx)))) ### All 37
% 1.40/1.63 39. (-. ((aElementOf0 T_1 (xS)) => (aElementOf0 T_1 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))))) (All W0, ((aSet0 W0) => (All W1, ((aElementOf0 W1 W0) => (aElement0 W1))))) (aSet0 (xS)) (All W0, ((aSet0 W0) => (All W1, ((aSubsetOf0 W1 W0) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 W0)))))))) (aSet0 (sdtpldt0 (xS) (xx))) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (-. (aElementOf0 (xx) (xS))) (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx)))))) ### NotImply 38
% 1.40/1.63 40. (-. (All W0, ((aElementOf0 W0 (xS)) => (aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx)))))) (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx)))))) (-. (aElementOf0 (xx) (xS))) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (aSet0 (sdtpldt0 (xS) (xx))) (All W0, ((aSet0 W0) => (All W1, ((aSubsetOf0 W1 W0) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 W0)))))))) (aSet0 (xS)) (All W0, ((aSet0 W0) => (All W1, ((aElementOf0 W1 W0) => (aElement0 W1))))) ### NotAllEx 39
% 1.40/1.63 41. (-. (((aSet0 (sdtpldt0 (xS) (xx))) /\ (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx))))))) => (((aSet0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) /\ (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx))))))) => ((All W0, ((aElementOf0 W0 (xS)) => (aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))))) \/ (aSubsetOf0 (xS) (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))))))) (All W0, ((aSet0 W0) => (All W1, ((aElementOf0 W1 W0) => (aElement0 W1))))) (aSet0 (xS)) (All W0, ((aSet0 W0) => (All W1, ((aSubsetOf0 W1 W0) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 W0)))))))) (-. (aElementOf0 (xx) (xS))) ### ConjTree 40
% 1.40/1.63 42. (aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) (-. (aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx)))) ### Axiom
% 1.40/1.63 43. (-. (aElementOf0 T_2 (sdtpldt0 (xS) (xx)))) (aElementOf0 T_2 (sdtpldt0 (xS) (xx))) ### Axiom
% 1.40/1.63 44. ((aElement0 T_2) /\ ((aElementOf0 T_2 (sdtpldt0 (xS) (xx))) /\ (T_2 != (xx)))) (-. (aElementOf0 T_2 (sdtpldt0 (xS) (xx)))) ### ConjTree 43
% 1.40/1.63 45. ((aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 T_2) /\ ((aElementOf0 T_2 (sdtpldt0 (xS) (xx))) /\ (T_2 != (xx))))) (-. (aElementOf0 T_2 (sdtpldt0 (xS) (xx)))) (aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) ### Equiv 42 44
% 1.40/1.63 46. (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx)))))) (aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) (-. (aElementOf0 T_2 (sdtpldt0 (xS) (xx)))) ### All 45
% 1.40/1.63 47. (-. (aElementOf0 T_2 (xS))) (aElementOf0 T_2 (xS)) ### Axiom
% 1.40/1.63 48. (aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) (-. (aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx)))) ### Axiom
% 1.40/1.63 49. (T_2 = (xx)) (T_2 != (xx)) ### Axiom
% 1.40/1.63 50. ((aElement0 T_2) /\ ((aElementOf0 T_2 (sdtpldt0 (xS) (xx))) /\ (T_2 != (xx)))) (T_2 = (xx)) ### ConjTree 49
% 1.40/1.63 51. ((aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 T_2) /\ ((aElementOf0 T_2 (sdtpldt0 (xS) (xx))) /\ (T_2 != (xx))))) (T_2 = (xx)) (aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) ### Equiv 48 50
% 1.40/1.63 52. (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx)))))) (aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) (T_2 = (xx)) ### All 51
% 1.40/1.63 53. ((aElementOf0 T_2 (xS)) \/ (T_2 = (xx))) (aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx)))))) (-. (aElementOf0 T_2 (xS))) ### Or 47 52
% 1.40/1.63 54. ((aElement0 T_2) /\ ((aElementOf0 T_2 (xS)) \/ (T_2 = (xx)))) (-. (aElementOf0 T_2 (xS))) (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx)))))) (aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) ### And 53
% 1.40/1.63 55. ((aElementOf0 T_2 (sdtpldt0 (xS) (xx))) <=> ((aElement0 T_2) /\ ((aElementOf0 T_2 (xS)) \/ (T_2 = (xx))))) (-. (aElementOf0 T_2 (xS))) (aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx)))))) ### Equiv 46 54
% 1.40/1.63 56. (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx)))))) (aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) (-. (aElementOf0 T_2 (xS))) ### All 55
% 1.40/1.63 57. (-. ((aElementOf0 T_2 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) => (aElementOf0 T_2 (xS)))) (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx)))))) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) ### NotImply 56
% 1.40/1.63 58. (-. (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) => (aElementOf0 W0 (xS))))) (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx)))))) (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx)))))) ### NotAllEx 57
% 1.40/1.63 59. (-. (((aSet0 (sdtpldt0 (xS) (xx))) /\ (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx))))))) => (((aSet0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) /\ (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx))))))) => ((All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) => (aElementOf0 W0 (xS)))) \/ (aSubsetOf0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx)) (xS)))))) ### ConjTree 58
% 1.40/1.63 60. (-. ((((aSet0 (sdtpldt0 (xS) (xx))) /\ (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx))))))) => (((aSet0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) /\ (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx))))))) => ((All W0, ((aElementOf0 W0 (xS)) => (aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))))) \/ (aSubsetOf0 (xS) (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx)))))) /\ (((aSet0 (sdtpldt0 (xS) (xx))) /\ (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx))))))) => (((aSet0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) /\ (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx))))))) => ((All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) => (aElementOf0 W0 (xS)))) \/ (aSubsetOf0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx)) (xS))))))) (-. (aElementOf0 (xx) (xS))) (All W0, ((aSet0 W0) => (All W1, ((aSubsetOf0 W1 W0) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 W0)))))))) (aSet0 (xS)) (All W0, ((aSet0 W0) => (All W1, ((aElementOf0 W1 W0) => (aElement0 W1))))) ### NotAnd 41 59
% 1.40/1.63 61. ((aElement0 (xx)) /\ (aSet0 (xS))) (All W0, ((aSet0 W0) => (All W1, ((aElementOf0 W1 W0) => (aElement0 W1))))) (All W0, ((aSet0 W0) => (All W1, ((aSubsetOf0 W1 W0) <=> ((aSet0 W1) /\ (All W2, ((aElementOf0 W2 W1) => (aElementOf0 W2 W0)))))))) (-. (aElementOf0 (xx) (xS))) (-. ((((aSet0 (sdtpldt0 (xS) (xx))) /\ (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx))))))) => (((aSet0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) /\ (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx))))))) => ((All W0, ((aElementOf0 W0 (xS)) => (aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))))) \/ (aSubsetOf0 (xS) (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx)))))) /\ (((aSet0 (sdtpldt0 (xS) (xx))) /\ (All W0, ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (xS)) \/ (W0 = (xx))))))) => (((aSet0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) /\ (All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) <=> ((aElement0 W0) /\ ((aElementOf0 W0 (sdtpldt0 (xS) (xx))) /\ (W0 != (xx))))))) => ((All W0, ((aElementOf0 W0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx))) => (aElementOf0 W0 (xS)))) \/ (aSubsetOf0 (sdtmndt0 (sdtpldt0 (xS) (xx)) (xx)) (xS))))))) ### And 60
% 1.40/1.63 % SZS output end Proof
% 1.40/1.63 (* END-PROOF *)
%------------------------------------------------------------------------------