Theorem nelne1 2414

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 2218	. . . 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 2367	. 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 1332   e. wcel 2125   =/= wne 2324

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 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1487  ax-17 1503  ax-ial 1511  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-cleq 2147  df-clel 2150  df-ne 2325

This theorem is referenced by:  elnelne1  2428  difsnb  3695