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Theorem ssnelpss 3683
Description: A subclass missing a member is a proper subclass. (Contributed by NM, 12-Jan-2002.)
Assertion
Ref Expression
ssnelpss  |-  ( A 
C_  B  ->  (
( C  e.  B  /\  -.  C  e.  A
)  ->  A  C.  B ) )

Proof of Theorem ssnelpss
StepHypRef Expression
1 nelneq2 2534 . . 3  |-  ( ( C  e.  B  /\  -.  C  e.  A
)  ->  -.  B  =  A )
2 eqcom 2437 . . 3  |-  ( B  =  A  <->  A  =  B )
31, 2sylnib 296 . 2  |-  ( ( C  e.  B  /\  -.  C  e.  A
)  ->  -.  A  =  B )
4 dfpss2 3424 . . 3  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
54baibr 873 . 2  |-  ( A 
C_  B  ->  ( -.  A  =  B  <->  A 
C.  B ) )
63, 5syl5ib 211 1  |-  ( A 
C_  B  ->  (
( C  e.  B  /\  -.  C  e.  A
)  ->  A  C.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3312    C. wpss 3313
This theorem is referenced by:  ssnelpssd  3684  canthp1lem2  8518  nqpr  8881  uzindi  11310  nthruc  12840  nthruz  12841  vitali  19495  onpsstopbas  26145
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-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-cleq 2428  df-clel 2431  df-ne 2600  df-pss 3328
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