Theorem nssne1 3055

Description: Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)

Assertion

Ref	Expression
nssne1	|- ( ( A C_ B /\ -. A C_ C ) -> B =/= C )

Proof of Theorem nssne1

Step	Hyp	Ref	Expression
1		sseq2 3021	. . . 4 |- ( B = C -> ( A C_ B <-> A C_ C ) )
2	1	biimpcd 157	. . 3 |- ( A C_ B -> ( B = C -> A C_ C ) )
3	2	necon3bd 2288	. 2 |- ( A C_ B -> ( -. A C_ C -> B =/= C ) )
4	3	imp 122	1 |- ( ( A C_ B /\ -. A C_ C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1284   =/= wne 2245   C_ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-ne 2246  df-in 2979  df-ss 2986

This theorem is referenced by: (None)