Theorem elpwid 3667

Description: An element of a power class is a subclass. Deduction form of elpwi 3665. (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 3665	. 2 |- ( A e. ~P B -> A C_ B )
3	1, 2	syl 14	1 |- ( ph -> A C_ B )

Syntax hints:   -> wi 4   e. wcel 2202   C_ wss 3201   ~Pcpw 3656

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-pw 3658

This theorem is referenced by:  fopwdom  7065  ssenen  7080  fival  7212  fiuni  7220  3nelsucpw1  7495  elnp1st2nd  7739  ixxssxr  10179  elfzoelz  10427  restid2  13394  epttop  14884  neiss2  14936  blssm  15215  blin2  15226  cncfrss  15369  cncfrss2  15370  dvidsslem  15487  dvconstss  15492  plybss  15527  uhgrss  15999  upgrss  16023  upgr1een  16048  usgrss  16101  eupth2lemsfi  16402  pw1ndom3lem  16692  pwle2  16703