Theorem elpwid 3632

Description: An element of a power class is a subclass. Deduction form of elpwi 3630. (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 3630	. 2 |- ( A e. ~P B -> A C_ B )
3	1, 2	syl 14	1 |- ( ph -> A C_ B )

Syntax hints:   -> wi 4   e. wcel 2177   C_ wss 3170   ~Pcpw 3621

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-ss 3183  df-pw 3623

This theorem is referenced by:  fopwdom  6948  ssenen  6963  fival  7087  fiuni  7095  3nelsucpw1  7365  elnp1st2nd  7609  ixxssxr  10042  elfzoelz  10289  restid2  13155  epttop  14637  neiss2  14689  blssm  14968  blin2  14979  cncfrss  15122  cncfrss2  15123  dvidsslem  15240  dvconstss  15245  plybss  15280  uhgrss  15746  upgrss  15770  pwle2  16076