Theorem nelneq 1561

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 1534	. . . 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 956   e. wcel 958

This theorem is referenced by:  disjne 2315  difsn 2464  suc11reg 4605  renepnft 5537  renemnft 5538  topnem 10507  fipfil 10563  fipfil2 10564  fipfil2OLD 10565  cnfilca 10583  cnfilcaOLD 10584

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-17 971  ax-4 973  ax-5o 975  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-cleq 1469  df-clel 1472

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