Theorem nelneq 1558

Description: A way of showing two classes are not equal.

Assertion

Ref	Expression
nelneq	|- ((A e. C /\ -. B e. C) -> -. A = B)

Proof of Theorem nelneq

Step	Hyp	Ref	Expression
1		eleq1 1531	. . . 4 |- (A = B -> (A e. C <-> B e. C))
2	1	biimpcd 155	. . 3 |- (A e. C -> (A = B -> B e. C))
3	2	con3d 95	. 2 |- (A e. C -> (-. B e. C -> -. A = B))
4	3	imp 350	1 |- ((A e. C /\ -. B e. C) -> -. A = B)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956

This theorem is referenced by:  disjne 2311  difsn 2460  suc11reg 4585  renepnft 5518  renemnft 5519  topnem 10430  fipfil 10474  fipfil2 10475  cnfilca 10487

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 961  ax-17 969  ax-4 971  ax-5o 973  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-cleq 1467  df-clel 1470

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