Theorem ssneld 3099

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 3096	. 2 |- ( ph -> ( C e. A -> C e. B ) )
3	2	con3d 620	1 |- ( ph -> ( -. C e. B -> -. C e. A ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1480   C_ wss 3071

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 603  ax-in2 604  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084

This theorem is referenced by:  ssneldd  3100  sumdc  11127  summodclem2a  11150  zsumdc  11153  isumss2  11162  decidin  13004