Theorem ssneldd 3150

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 3149	. 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 2141   C_ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134

This theorem is referenced by:  0nelrel  4657  addnqprlemfl  7521  addnqprlemfu  7522  mulnqprlemfl  7537  mulnqprlemfu  7538  cauappcvgprlemladdru  7618  fprodntrivap  11547  fprodssdc  11553