Theorem nelne1 2430

Description: Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)

Assertion

Ref	Expression
nelne1	|- ( ( A e. B /\ -. A e. C ) -> B =/= C )

Proof of Theorem nelne1

Step	Hyp	Ref	Expression
1		eleq2 2234	. . . 4 |- ( B = C -> ( A e. B <-> A e. C ) )
2	1	biimpcd 158	. . 3 |- ( A e. B -> ( B = C -> A e. C ) )
3	2	necon3bd 2383	. 2 |- ( A e. B -> ( -. A e. C -> B =/= C ) )
4	3	imp 123	1 |- ( ( A e. B /\ -. A e. C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   =/= wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166  df-ne 2341

This theorem is referenced by:  elnelne1  2444  difsnb  3723