Theorem nelne1 2490

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 2293	. . . 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 2443	. 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 1395   e. wcel 2200   =/= wne 2400

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 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-ne 2401

This theorem is referenced by:  elnelne1  2504  difsnb  3811