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Theorem ssnelpss 3634
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 2486 . . 3  |-  ( ( C  e.  B  /\  -.  C  e.  A
)  ->  -.  B  =  A )
2 eqcom 2389 . . 3  |-  ( B  =  A  <->  A  =  B )
31, 2sylnib 296 . 2  |-  ( ( C  e.  B  /\  -.  C  e.  A
)  ->  -.  A  =  B )
4 dfpss2 3375 . . 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 1649    e. wcel 1717    C_ wss 3263    C. wpss 3264
This theorem is referenced by:  ssnelpssd  3635  canthp1lem2  8461  nqpr  8824  uzindi  11247  nthruc  12777  nthruz  12778  vitali  19372  onpsstopbas  25894
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-11 1753  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-cleq 2380  df-clel 2383  df-ne 2552  df-pss 3279
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