Theorem elpwid 3660

Description: An element of a power class is a subclass. Deduction form of elpwi 3658. (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 3658	. 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 3197   ~Pcpw 3649

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 2801  df-in 3203  df-ss 3210  df-pw 3651

This theorem is referenced by:  fopwdom  7005  ssenen  7020  fival  7148  fiuni  7156  3nelsucpw1  7430  elnp1st2nd  7674  ixxssxr  10108  elfzoelz  10355  restid2  13296  epttop  14779  neiss2  14831  blssm  15110  blin2  15121  cncfrss  15264  cncfrss2  15265  dvidsslem  15382  dvconstss  15387  plybss  15422  uhgrss  15890  upgrss  15914  usgrss  15990  pw1ndom3lem  16412  pwle2  16423