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Theorem elpwg 3437
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 2150 . 2 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵𝐴 ∈ 𝒫 𝐵))
2 sseq1 3047 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 vex 2622 . . 3 𝑥 ∈ V
43elpw 3435 . 2 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
51, 2, 4vtoclbg 2680 1 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wcel 1438  wss 2999  𝒫 cpw 3429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-pw 3431
This theorem is referenced by:  elpwi  3438  elpwb  3439  pwidg  3443  prsspwg  3596  elpw2g  3992  snelpwi  4039  prelpwi  4041  pwel  4045  eldifpw  4299  f1opw2  5850  2pwuninelg  6048  tfrlemibfn  6093  tfr1onlembfn  6109  tfrcllembfn  6122  elpmg  6421  fopwdom  6552  fiinopn  11601
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