Theorem nssne2 3242

Description: Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)

Assertion

Ref	Expression
nssne2	|- ( ( A C_ C /\ -. B C_ C ) -> A =/= B )

Proof of Theorem nssne2

Step	Hyp	Ref	Expression
1		sseq1 3206	. . . 4 |- ( A = B -> ( A C_ C <-> B C_ C ) )
2	1	biimpcd 159	. . 3 |- ( A C_ C -> ( A = B -> B C_ C ) )
3	2	necon3bd 2410	. 2 |- ( A C_ C -> ( -. B C_ C -> A =/= B ) )
4	3	imp 124	1 |- ( ( A C_ C /\ -. B C_ C ) -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   =/= wne 2367   C_ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ne 2368  df-in 3163  df-ss 3170

This theorem is referenced by: (None)