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Theorem elpwg 3513
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)
Assertion
Ref Expression
elpwg (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Proof of Theorem elpwg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2200 . 2 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵𝐴 ∈ 𝒫 𝐵))
2 sseq1 3115 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 vex 2684 . . 3 𝑥 ∈ V
43elpw 3511 . 2 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
51, 2, 4vtoclbg 2742 1 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  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:  elpwi  3514  elpwb  3515  pwidg  3519  prsspwg  3674  elpw2g  4076  snelpwi  4129  prelpwi  4131  pwel  4135  eldifpw  4393  f1opw2  5969  2pwuninelg  6173  tfrlemibfn  6218  tfr1onlembfn  6234  tfrcllembfn  6247  elpmg  6551  fopwdom  6723  fiinopn  12160  ssntr  12280
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