Theorem elpwid 3663

Description: An element of a power class is a subclass. Deduction form of elpwi 3661. (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 3661	. 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 3200   ~Pcpw 3652

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by:  fopwdom  7022  ssenen  7037  fival  7169  fiuni  7177  3nelsucpw1  7452  elnp1st2nd  7696  ixxssxr  10135  elfzoelz  10382  restid2  13349  epttop  14833  neiss2  14885  blssm  15164  blin2  15175  cncfrss  15318  cncfrss2  15319  dvidsslem  15436  dvconstss  15441  plybss  15476  uhgrss  15945  upgrss  15969  upgr1een  15994  usgrss  16047  eupth2lemsfi  16348  pw1ndom3lem  16639  pwle2  16650