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Theorem elpwid 3634
Description: An element of a power class is a subclass. Deduction form of elpwi 3633. (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
StepHypRef Expression
1 elpwid.1 . 2  |-  ( ph  ->  A  e.  ~P B
)
2 elpwi 3633 . 2  |-  ( A  e.  ~P B  ->  A  C_  B )
31, 2syl 15 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    C_ wss 3152   ~Pcpw 3625
This theorem is referenced by:  incexclem  12295  mreexexlem4d  13549  mreexexd  13550  mreexdomd  13551  acsfiindd  14280  lssacsex  15897  br2base  23574  indf1ofs  23609  usgrass  28110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627
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