Theorem ssneldd 3227

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 3226	. 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 2200   C_ wss 3197

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 617  ax-in2 618  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by:  0nelrel  4764  addnqprlemfl  7742  addnqprlemfu  7743  mulnqprlemfl  7758  mulnqprlemfu  7759  cauappcvgprlemladdru  7839  fprodntrivap  12090  fprodssdc  12096