Theorem ssnelpssd 3614

Description: Subclass inclusion with one element of the superclass missing is proper subclass inclusion. Deduction form of ssnelpss 3613. (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 3613	. . 3 |- (A C_ B -> ((C e. B /\ -. C e. A) -> A C. B))
5	3, 4	syl 15	. 2 |- (ph -> ((C e. B /\ -. C e. A) -> A C. B))
6	1, 2, 5	mp2and 660	1 |- (ph -> A C. B)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   e. wcel 1710   C_ wss 3257   C. wpss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-ne 2518  df-pss 3261

This theorem is referenced by: (None)