Theorem nssne1 3251

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 3217	. . . 4 |- ( B = C -> ( A C_ B <-> A C_ C ) )
2	1	biimpcd 159	. . 3 |- ( A C_ B -> ( B = C -> A C_ C ) )
3	2	necon3bd 2419	. 2 |- ( A C_ B -> ( -. A C_ C -> B =/= C ) )
4	3	imp 124	1 |- ( ( A C_ B /\ -. A C_ C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1373   =/= wne 2376   C_ wss 3166

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 615  ax-in2 616  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-ne 2377  df-in 3172  df-ss 3179

This theorem is referenced by: (None)