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Theorem elpwg 3488
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 2180 . 2 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵𝐴 ∈ 𝒫 𝐵))
2 sseq1 3090 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 vex 2663 . . 3 𝑥 ∈ V
43elpw 3486 . 2 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
51, 2, 4vtoclbg 2721 1 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1465  wss 3041  𝒫 cpw 3480
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482
This theorem is referenced by:  elpwi  3489  elpwb  3490  pwidg  3494  prsspwg  3649  elpw2g  4051  snelpwi  4104  prelpwi  4106  pwel  4110  eldifpw  4368  f1opw2  5944  2pwuninelg  6148  tfrlemibfn  6193  tfr1onlembfn  6209  tfrcllembfn  6222  elpmg  6526  fopwdom  6698  fiinopn  12098  ssntr  12218
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