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Theorem elpwid 3485
Description: An element of a power class is a subclass. Deduction form of elpwi 3483. (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 3483 . 2  |-  ( A  e.  ~P B  ->  A  C_  B )
31, 2syl 14 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1461    C_ wss 3035   ~Pcpw 3474
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-in 3041  df-ss 3048  df-pw 3476
This theorem is referenced by:  fopwdom  6681  ssenen  6696  fival  6808  fiuni  6816  elnp1st2nd  7226  ixxssxr  9570  elfzoelz  9811  restid2  11966  epttop  12096  neiss2  12148  blssm  12404  blin2  12415  cncfrss  12542  cncfrss2  12543  pwle2  12876
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