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Theorem elpwg 3367
 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 2100 . 2 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵𝐴 ∈ 𝒫 𝐵))
2 sseq1 2966 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 vex 2560 . . 3 𝑥 ∈ V
43elpw 3365 . 2 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
51, 2, 4vtoclbg 2614 1 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1393   ⊆ wss 2917  𝒫 cpw 3359 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361 This theorem is referenced by:  elpwi  3368  pwidg  3372  prsspwg  3523  elpw2g  3910  snelpwi  3948  prelpwi  3950  pwel  3954  eldifpw  4208  f1opw2  5706  2pwuninelg  5898  tfrlemibfn  5942  fopwdom  6310
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