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Theorem velpw 3512
Description: Setvar variable membership in a power class (common case). See elpw 3511. (Contributed by David A. Wheeler, 8-Dec-2018.)
Assertion
Ref Expression
velpw (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem velpw
StepHypRef Expression
1 vex 2684 . 2 𝑥 ∈ V
21elpw 3511 1 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 1480  wss 3066  𝒫 cpw 3505
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507
This theorem is referenced by:  ordpwsucss  4477  fabexg  5305  abexssex  6016  qsss  6481  mapval2  6565  pmsspw  6570  uniixp  6608  exmidpw  6795  fival  6851  npsspw  7272  restsspw  12119  istopon  12169  isbasis2g  12201  tgval2  12209  unitg  12220  distop  12243  cldss2  12264  ntreq0  12290  discld  12294  neisspw  12306  restdis  12342  cnntr  12383
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