Theorem ssnelpssd 3719

Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3718. (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

Step	Hyp	Ref	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 3718	. . 3 |- ( A C_ B -> ( ( C e. B /\ -. C e. A ) -> A C. B ) )
5	3, 4	syl 17	. 2 |- ( ph -> ( ( C e. B /\ -. C e. A ) -> A C. B ) )
6	1, 2, 5	mp2and 715	1 |- ( ph -> A C. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   e. wcel 1990   C_ wss 3574   C. wpss 3575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-ne 2795  df-pss 3590

This theorem is referenced by:  isfin4-3  9137  canth4  9469  mrieqv2d  16299  symggen  17890  pgpfac1lem1  18473  pgpfaclem2  18481