Theorem ssneldd 3182

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 3181	. 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 2164   C_ wss 3153

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166

This theorem is referenced by:  0nelrel  4705  addnqprlemfl  7619  addnqprlemfu  7620  mulnqprlemfl  7635  mulnqprlemfu  7636  cauappcvgprlemladdru  7716  fprodntrivap  11727  fprodssdc  11733