Theorem elpwid 3516

Description: An element of a power class is a subclass. Deduction form of elpwi 3514. (Contributed by David Moews, 1-May-2017.)

Hypothesis

Ref	Expression
elpwid.1	|- ( ph -> A e. ~P B )

Assertion

Ref	Expression
elpwid	|- ( ph -> A C_ B )

Proof of Theorem elpwid

Step	Hyp	Ref	Expression
1		elpwid.1	. 2 |- ( ph -> A e. ~P B )
2		elpwi 3514	. 2 |- ( A e. ~P B -> A C_ B )
3	1, 2	syl 14	1 |- ( ph -> A C_ B )

Syntax hints:   -> wi 4   e. wcel 1480   C_ wss 3066   ~Pcpw 3505

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507

This theorem is referenced by:  fopwdom  6723  ssenen  6738  fival  6851  fiuni  6859  elnp1st2nd  7277  ixxssxr  9676  elfzoelz  9917  restid2  12118  epttop  12248  neiss2  12300  blssm  12579  blin2  12590  cncfrss  12720  cncfrss2  12721  pwle2  13182