Theorem elpwid 3570

Description: An element of a power class is a subclass. Deduction form of elpwi 3568. (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 3568	. 2 |- ( A e. ~P B -> A C_ B )
3	1, 2	syl 14	1 |- ( ph -> A C_ B )

Syntax hints:   -> wi 4   e. wcel 2136   C_ wss 3116   ~Pcpw 3559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561

This theorem is referenced by:  fopwdom  6802  ssenen  6817  fival  6935  fiuni  6943  3nelsucpw1  7190  elnp1st2nd  7417  ixxssxr  9836  elfzoelz  10082  restid2  12565  epttop  12740  neiss2  12792  blssm  13071  blin2  13082  cncfrss  13212  cncfrss2  13213  pwle2  13888