Theorem nssne2 3662

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 3626	. . . 4 |- ( A = B -> ( A C_ C <-> B C_ C ) )
2	1	biimpcd 239	. . 3 |- ( A C_ C -> ( A = B -> B C_ C ) )
3	2	necon3bd 2808	. 2 |- ( A C_ C -> ( -. B C_ C -> A =/= B ) )
4	3	imp 445	1 |- ( ( A C_ C /\ -. B C_ C ) -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   =/= wne 2794   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ne 2795  df-in 3581  df-ss 3588

This theorem is referenced by:  atcvatlem  29244  mdsymlem3  29264  disjdifprg  29388  mapdh6aN  37024  mapdh8e  37073  hdmap1l6a  37099