MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nssne2 Unicode version

Theorem nssne2 3248
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
StepHypRef Expression
1 sseq1 3212 . . . 4  |-  ( A  =  B  ->  ( A  C_  C  <->  B  C_  C
) )
21biimpcd 215 . . 3  |-  ( A 
C_  C  ->  ( A  =  B  ->  B 
C_  C ) )
32necon3bd 2496 . 2  |-  ( A 
C_  C  ->  ( -.  B  C_  C  ->  A  =/=  B ) )
43imp 418 1  |-  ( ( A  C_  C  /\  -.  B  C_  C )  ->  A  =/=  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    =/= wne 2459    C_ wss 3165
This theorem is referenced by:  atcvatlem  22981  mdsymlem3  23001  disjdifprg  23367  mapdh6aN  32547  mapdh8e  32596  hdmap1l6a  32622
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-ne 2461  df-in 3172  df-ss 3179
  Copyright terms: Public domain W3C validator