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Theorem ssnelpss 3289
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 2139 . . 3  |-  ( ( C  e.  B  /\  -.  C  e.  A
)  ->  -.  B  =  A )
2 eqcom 2042 . . 3  |-  ( B  =  A  <->  A  =  B )
31, 2sylnib 601 . 2  |-  ( ( C  e.  B  /\  -.  C  e.  A
)  ->  -.  A  =  B )
4 dfpss2 3029 . . 3  |-  ( A 
C.  B  <->  ( A  C_  B  /\  -.  A  =  B ) )
54baibr 829 . 2  |-  ( A 
C_  B  ->  ( -.  A  =  B  <->  A 
C.  B ) )
63, 5syl5ib 143 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 97    = wceq 1243    e. wcel 1393    C_ wss 2917    C. wpss 2918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-ne 2206  df-pss 2933
This theorem is referenced by:  ssnelpssd  3290
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