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Theorem nssne1 3064
Description: Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
Assertion
Ref Expression
nssne1  |-  ( ( A  C_  B  /\  -.  A  C_  C )  ->  B  =/=  C
)

Proof of Theorem nssne1
StepHypRef Expression
1 sseq2 3030 . . . 4  |-  ( B  =  C  ->  ( A  C_  B  <->  A  C_  C
) )
21biimpcd 157 . . 3  |-  ( A 
C_  B  ->  ( B  =  C  ->  A 
C_  C ) )
32necon3bd 2292 . 2  |-  ( A 
C_  B  ->  ( -.  A  C_  C  ->  B  =/=  C ) )
43imp 122 1  |-  ( ( A  C_  B  /\  -.  A  C_  C )  ->  B  =/=  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1285    =/= wne 2249    C_ wss 2982
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 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-ne 2250  df-in 2988  df-ss 2995
This theorem is referenced by: (None)
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