Theorem ssnelpssd 3290

Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3289. (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 3289	. . 3 |- ( A C_ B -> ( ( C e. B /\ -. C e. A ) -> A C. B ) )
5	3, 4	syl 14	. 2 |- ( ph -> ( ( C e. B /\ -. C e. A ) -> A C. B ) )
6	1, 2, 5	mp2and 409	1 |- ( ph -> A C. B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   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: (None)