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Theorem ssnelpssd 3692
Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3691. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ssnelpssd.1  |-  ( ph  ->  A  C_  B )
ssnelpssd.2  |-  ( ph  ->  C  e.  B )
ssnelpssd.3  |-  ( ph  ->  -.  C  e.  A
)
Assertion
Ref Expression
ssnelpssd  |-  ( ph  ->  A  C.  B )

Proof of Theorem ssnelpssd
StepHypRef Expression
1 ssnelpssd.2 . 2  |-  ( ph  ->  C  e.  B )
2 ssnelpssd.3 . 2  |-  ( ph  ->  -.  C  e.  A
)
3 ssnelpssd.1 . . 3  |-  ( ph  ->  A  C_  B )
4 ssnelpss 3691 . . 3  |-  ( A 
C_  B  ->  (
( C  e.  B  /\  -.  C  e.  A
)  ->  A  C.  B ) )
53, 4syl 16 . 2  |-  ( ph  ->  ( ( C  e.  B  /\  -.  C  e.  A )  ->  A  C.  B ) )
61, 2, 5mp2and 661 1  |-  ( ph  ->  A  C.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1725    C_ wss 3320    C. wpss 3321
This theorem is referenced by:  isfin4-3  8195  canth4  8522  mrieqv2d  13864  pgpfac1lem1  15632  pgpfaclem2  15640  symggen  27388
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 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-cleq 2429  df-clel 2432  df-ne 2601  df-pss 3336
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