Theorem ssneldd 3195

Description: If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
ssneld.1	|- ( ph -> A C_ B )
ssneldd.2	|- ( ph -> -. C e. B )

Assertion

Ref	Expression
ssneldd	|- ( ph -> -. C e. A )

Proof of Theorem ssneldd

Step	Hyp	Ref	Expression
1		ssneldd.2	. 2 |- ( ph -> -. C e. B )
2		ssneld.1	. . 3 |- ( ph -> A C_ B )
3	2	ssneld 3194	. 2 |- ( ph -> ( -. C e. B -> -. C e. A ) )
4	1, 3	mpd 13	1 |- ( ph -> -. C e. A )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2175   C_ wss 3165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-in 3171  df-ss 3178

This theorem is referenced by:  0nelrel  4720  addnqprlemfl  7671  addnqprlemfu  7672  mulnqprlemfl  7687  mulnqprlemfu  7688  cauappcvgprlemladdru  7768  fprodntrivap  11866  fprodssdc  11872