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Theorem ssnelpssd 3531
Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3530. (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 3530 . . 3  |-  ( A 
C_  B  ->  (
( C  e.  B  /\  -.  C  e.  A
)  ->  A  C.  B ) )
53, 4syl 15 . 2  |-  ( ph  ->  ( ( C  e.  B  /\  -.  C  e.  A )  ->  A  C.  B ) )
61, 2, 5mp2and 660 1  |-  ( ph  ->  A  C.  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    e. wcel 1696    C_ wss 3165    C. wpss 3166
This theorem is referenced by:  mrieqv2d  13557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-cleq 2289  df-clel 2292  df-ne 2461  df-pss 3181
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