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Theorem velpw 3561
Description: Setvar variable membership in a power class (common case). See elpw 3560. (Contributed by David A. Wheeler, 8-Dec-2018.)
Assertion
Ref Expression
velpw  |-  ( x  e.  ~P A  <->  x  C_  A
)
Distinct variable group:    x, A

Proof of Theorem velpw
StepHypRef Expression
1 vex 2725 . 2  |-  x  e. 
_V
21elpw 3560 1  |-  ( x  e.  ~P A  <->  x  C_  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 2135    C_ wss 3112   ~Pcpw 3554
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2724  df-in 3118  df-ss 3125  df-pw 3556
This theorem is referenced by:  ordpwsucss  4539  fabexg  5370  abexssex  6086  qsss  6552  mapval2  6636  pmsspw  6641  uniixp  6679  exmidpw  6866  exmidpweq  6867  pw1fin  6868  pw1dc0el  6869  fival  6927  npsspw  7404  restsspw  12528  istopon  12578  isbasis2g  12610  tgval2  12618  unitg  12629  distop  12652  cldss2  12673  ntreq0  12699  discld  12703  neisspw  12715  restdis  12751  cnntr  12792
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