Theorem nssne1 3150

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 3116	. . . 4 |- ( B = C -> ( A C_ B <-> A C_ C ) )
2	1	biimpcd 158	. . 3 |- ( A C_ B -> ( B = C -> A C_ C ) )
3	2	necon3bd 2349	. 2 |- ( A C_ B -> ( -. A C_ C -> B =/= C ) )
4	3	imp 123	1 |- ( ( A C_ B /\ -. A C_ C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1331   =/= wne 2306   C_ wss 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ne 2307  df-in 3072  df-ss 3079

This theorem is referenced by: (None)