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Theorem nssne1 3396
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 3362 . . . 4  |-  ( B  =  C  ->  ( A  C_  B  <->  A  C_  C
) )
21biimpcd 216 . . 3  |-  ( A 
C_  B  ->  ( B  =  C  ->  A 
C_  C ) )
32necon3bd 2635 . 2  |-  ( A 
C_  B  ->  ( -.  A  C_  C  ->  B  =/=  C ) )
43imp 419 1  |-  ( ( A  C_  B  /\  -.  A  C_  C )  ->  B  =/=  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    =/= wne 2598    C_ wss 3312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-ne 2600  df-in 3319  df-ss 3326
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