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Theorem ssnelpssd 3740
Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3739. (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 3739 . . 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 674 1  |-  ( ph  ->  A  C.  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1761    C_ wss 3325    C. wpss 3326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-12 1797  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-cleq 2434  df-clel 2437  df-ne 2606  df-pss 3341
This theorem is referenced by:  isfin4-3  8480  canth4  8810  mrieqv2d  14573  symggen  15969  pgpfac1lem1  16565  pgpfaclem2  16573
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