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Theorem nssne1 3247
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 3213 . . . 4  |-  ( B  =  C  ->  ( A  C_  B  <->  A  C_  C
) )
21biimpcd 215 . . 3  |-  ( A 
C_  B  ->  ( B  =  C  ->  A 
C_  C ) )
32necon3bd 2496 . 2  |-  ( A 
C_  B  ->  ( -.  A  C_  C  ->  B  =/=  C ) )
43imp 418 1  |-  ( ( A  C_  B  /\  -.  A  C_  C )  ->  B  =/=  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    =/= wne 2459    C_ wss 3165
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
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