Theorem nelne1 2335

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 2142	. . . 4 |- ( B = C -> ( A e. B <-> A e. C ) )
2	1	biimpcd 157	. . 3 |- ( A e. B -> ( B = C -> A e. C ) )
3	2	necon3bd 2288	. 2 |- ( A e. B -> ( -. A e. C -> B =/= C ) )
4	3	imp 122	1 |- ( ( A e. B /\ -. A e. C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   =/= wne 2245

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077  df-ne 2246

This theorem is referenced by:  difsnb  3528