Theorem ssneld 3144

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 3141	. 2 |- ( ph -> ( C e. A -> C e. B ) )
3	2	con3d 621	1 |- ( ph -> ( -. C e. B -> -. C e. A ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 2136   C_ wss 3116

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 604  ax-in2 605  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by:  ssneldd  3145  sumdc  11299  summodclem2a  11322  zsumdc  11325  isumss2  11334  zproddc  11520  prodssdc  11530  decidin  13688