Theorem ssneld 3203

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

Hypothesis

Ref	Expression
ssneld.1	|- ( ph -> A C_ B )

Assertion

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

Proof of Theorem ssneld

Step	Hyp	Ref	Expression
1		ssneld.1	. . 3 |- ( ph -> A C_ B )
2	1	sseld 3200	. 2 |- ( ph -> ( C e. A -> C e. B ) )
3	2	con3d 632	1 |- ( ph -> ( -. C e. B -> -. C e. A ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2178   C_ wss 3174

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 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187

This theorem is referenced by:  ssneldd  3204  sumdc  11784  summodclem2a  11807  zsumdc  11810  isumss2  11819  zproddc  12005  prodssdc  12015  decidin  15933