Theorem nelne1 2295

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 2101	. . . 4 |- ( B = C -> ( A e. B <-> A e. C ) )
2	1	biimpcd 148	. . 3 |- ( A e. B -> ( B = C -> A e. C ) )
3	2	necon3bd 2248	. 2 |- ( A e. B -> ( -. A e. C -> B =/= C ) )
4	3	imp 115	1 |- ( ( A e. B /\ -. A e. C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   =/= wne 2204

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-ne 2206

This theorem is referenced by:  difsnb  3506