ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nssne2 Unicode version

Theorem nssne2 3057
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 3021 . . . 4  |-  ( A  =  B  ->  ( A  C_  C  <->  B  C_  C
) )
21biimpcd 157 . . 3  |-  ( A 
C_  C  ->  ( A  =  B  ->  B 
C_  C ) )
32necon3bd 2289 . 2  |-  ( A 
C_  C  ->  ( -.  B  C_  C  ->  A  =/=  B ) )
43imp 122 1  |-  ( ( A  C_  C  /\  -.  B  C_  C )  ->  A  =/=  B
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1285    =/= wne 2246    C_ wss 2974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-ne 2247  df-in 2980  df-ss 2987
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator