Theorem nssne2 3161

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 3125	. . . 4 |- ( A = B -> ( A C_ C <-> B C_ C ) )
2	1	biimpcd 158	. . 3 |- ( A C_ C -> ( A = B -> B C_ C ) )
3	2	necon3bd 2352	. 2 |- ( A C_ C -> ( -. B C_ C -> A =/= B ) )
4	3	imp 123	1 |- ( ( A C_ C /\ -. B C_ C ) -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1332   =/= wne 2309   C_ wss 3076

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ne 2310  df-in 3082  df-ss 3089

This theorem is referenced by: (None)