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Theorem ssnelpss 3859
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 2535 . . 3  |-  ( ( C  e.  B  /\  -.  C  e.  A
)  ->  -.  B  =  A )
2 eqcom 2431 . . 3  |-  ( B  =  A  <->  A  =  B )
31, 2sylnib 305 . 2  |-  ( ( C  e.  B  /\  -.  C  e.  A
)  ->  -.  A  =  B )
4 dfpss2 3550 . . 3  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
54baibr 912 . 2  |-  ( A 
C_  B  ->  ( -.  A  =  B  <->  A 
C.  B ) )
63, 5syl5ib 222 1  |-  ( A 
C_  B  ->  (
( C  e.  B  /\  -.  C  e.  A
)  ->  A  C.  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    C_ wss 3436    C. wpss 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-cleq 2414  df-clel 2417  df-ne 2616  df-pss 3452
This theorem is referenced by:  ssnelpssd  3860  ssexnelpss  3861  canthp1lem2  9085  nqpr  9446  uzindi  12200  nthruc  14302  nthruz  14303  vitali  22569  onpsstopbas  31095
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