Theorem nelne1 2437

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 2241	. . . 4 |- ( B = C -> ( A e. B <-> A e. C ) )
2	1	biimpcd 159	. . 3 |- ( A e. B -> ( B = C -> A e. C ) )
3	2	necon3bd 2390	. 2 |- ( A e. B -> ( -. A e. C -> B =/= C ) )
4	3	imp 124	1 |- ( ( A e. B /\ -. A e. C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   =/= wne 2347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-ne 2348

This theorem is referenced by:  elnelne1  2451  difsnb  3735