Theorem nelne2 2511

Description: Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)

Assertion

Ref	Expression
nelne2	|- ( ( A e. C /\ -. B e. C ) -> A =/= B )

Proof of Theorem nelne2

Step	Hyp	Ref	Expression
1		eleq1 2301	. . . 4 |- ( A = B -> ( A e. C <-> B e. C ) )
2	1	biimpcd 159	. . 3 |- ( A e. C -> ( A = B -> B e. C ) )
3	2	necon3bd 2463	. 2 |- ( A e. C -> ( -. B e. C -> A =/= B ) )
4	3	imp 124	1 |- ( ( A e. C /\ -. B e. C ) -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420

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 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-cleq 2231  df-clel 2234  df-ne 2421

This theorem is referenced by:  nelelne  2512  elnelne2  2525  zgt1rpn0n1  10075  cats1un  11471