Theorem nssne1 3186

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 3152	. . . 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 2370	. 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 1335   =/= wne 2327   C_ wss 3102

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 604  ax-in2 605  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-ne 2328  df-in 3108  df-ss 3115

This theorem is referenced by: (None)