Theorem nelneq2 2139

Description: A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)

Assertion

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

Proof of Theorem nelneq2

Step	Hyp	Ref	Expression
1		eleq2 2101	. . 3 |- ( B = C -> ( A e. B <-> A e. C ) )
2	1	biimpcd 148	. 2 |- ( A e. B -> ( B = C -> A e. C ) )
3	2	con3and 564	1 |- ( ( A e. B /\ -. A e. C ) -> -. B = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036

This theorem is referenced by:  ssnelpss  3289  dtruarb  3942  fzneuz  8963