Theorem nelneq2 1559

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

Assertion

Ref	Expression
nelneq2	|- ((A e. B /\ -. A e. C) -> -. B = C)

Proof of Theorem nelneq2

Step	Hyp	Ref	Expression
1		eleq2 1532	. . . 4 |- (B = C -> (A e. B <-> A e. C))
2	1	biimpcd 155	. . 3 |- (A e. B -> (B = C -> A e. C))
3	2	con3d 95	. 2 |- (A e. B -> (-. A e. C -> -. B = C))
4	3	imp 350	1 |- ((A e. B /\ -. A e. C) -> -. B = C)

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

This theorem is referenced by:  ssnelpss 2326  opth2 2795  opthwiener 2802  fzneuzt 6458

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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