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Theorem nssne1 3349
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 3315 . . . 4  |-  ( B  =  C  ->  ( A  C_  B  <->  A  C_  C
) )
21biimpcd 216 . . 3  |-  ( A 
C_  B  ->  ( B  =  C  ->  A 
C_  C ) )
32necon3bd 2589 . 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 1649    =/= wne 2552    C_ wss 3265
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-ne 2554  df-in 3272  df-ss 3279
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