Theorem elpwid 3685

Description: An element of a power class is a subclass. Deduction form of elpwi 3683. (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 3683	. 2 |- ( A e. ~P B -> A C_ B )
3	1, 2	syl 14	1 |- ( ph -> A C_ B )

Syntax hints:   -> wi 4   e. wcel 2205   C_ wss 3214   ~Pcpw 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-ss 3227  df-pw 3676

This theorem is referenced by:  fopwdom  7102  ssenen  7118  fival  7270  fiuni  7278  3nelsucpw1  7557  elnp1st2nd  7807  ixxssxr  10252  elfzoelz  10503  ballotfilem2  13172  ballotfilemfmpn  13178  restid2  13545  epttop  15081  neiss2  15133  blssm  15412  blin2  15423  cncfrss  15566  cncfrss2  15567  dvidsslem  15684  dvconstss  15689  plybss  15724  uhgrss  16196  upgrss  16220  upgr1een  16245  usgrss  16298  eupth2lemsfi  16599  pw1ndom3lem  16889  pwle2  16898