Theorem elpwid 3661

Description: An element of a power class is a subclass. Deduction form of elpwi 3659. (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 3659	. 2 |- ( A e. ~P B -> A C_ B )
3	1, 2	syl 14	1 |- ( ph -> A C_ B )

Syntax hints:   -> wi 4   e. wcel 2200   C_ wss 3198   ~Pcpw 3650

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-in 3204  df-ss 3211  df-pw 3652

This theorem is referenced by:  fopwdom  7017  ssenen  7032  fival  7160  fiuni  7168  3nelsucpw1  7442  elnp1st2nd  7686  ixxssxr  10125  elfzoelz  10372  restid2  13321  epttop  14804  neiss2  14856  blssm  15135  blin2  15146  cncfrss  15289  cncfrss2  15290  dvidsslem  15407  dvconstss  15412  plybss  15447  uhgrss  15916  upgrss  15940  usgrss  16016  pw1ndom3lem  16524  pwle2  16535