Theorem ssneldd 3200

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 3199	. 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 2177   C_ wss 3170

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-in 3176  df-ss 3183

This theorem is referenced by:  0nelrel  4734  addnqprlemfl  7702  addnqprlemfu  7703  mulnqprlemfl  7718  mulnqprlemfu  7719  cauappcvgprlemladdru  7799  fprodntrivap  11980  fprodssdc  11986