Theorem ssneld 3250

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 3247	. 2 |- ( ph -> ( C e. A -> C e. B ) )
3	2	con3d 640	1 |- ( ph -> ( -. C e. B -> -. C e. A ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2209   C_ wss 3220

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 623  ax-in2 624  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233

This theorem is referenced by:  ssneldd  3251  hashfibclem  11260  sumdc  12102  summodclem2a  12126  zsumdc  12129  isumss2  12138  zproddc  12324  prodssdc  12334  decidin  16739