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Theorem ssnelpss 3517
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 2382 . . 3  |-  ( ( C  e.  B  /\  -.  C  e.  A
)  ->  -.  B  =  A )
2 eqcom 2285 . . 3  |-  ( B  =  A  <->  A  =  B )
31, 2sylnib 295 . 2  |-  ( ( C  e.  B  /\  -.  C  e.  A
)  ->  -.  A  =  B )
4 dfpss2 3261 . . 3  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
54baibr 872 . 2  |-  ( A 
C_  B  ->  ( -.  A  =  B  <->  A 
C.  B ) )
63, 5syl5ib 210 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 358    = wceq 1623    e. wcel 1684    C_ wss 3152    C. wpss 3153
This theorem is referenced by:  ssnelpssd  3518  isfin4-3  7941  canthp1lem2  8275  nqpr  8638  uzindi  11043  nthruc  12529  nthruz  12530  pgpfac1lem1  15309  pgpfaclem2  15317  vitali  18968  onpsstopbas  24869  symggen  27411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-cleq 2276  df-clel 2279  df-ne 2448  df-pss 3168
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