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

Theorem nssne2 3527
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 3491 . . . 4  |-  ( A  =  B  ->  ( A  C_  C  <->  B  C_  C
) )
21biimpcd 227 . . 3  |-  ( A 
C_  C  ->  ( A  =  B  ->  B 
C_  C ) )
32necon3bd 2643 . 2  |-  ( A 
C_  C  ->  ( -.  B  C_  C  ->  A  =/=  B ) )
43imp 430 1  |-  ( ( A  C_  C  /\  -.  B  C_  C )  ->  A  =/=  B
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    =/= wne 2625    C_ wss 3442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-ne 2627  df-in 3449  df-ss 3456
This theorem is referenced by:  atcvatlem  27873  mdsymlem3  27893  disjdifprg  28024  mapdh6aN  35015  mapdh8e  35064  hdmap1l6a  35090
  Copyright terms: Public domain W3C validator