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Theorem elpw 3435
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
Hypothesis
Ref Expression
elpw.1 𝐴 ∈ V
Assertion
Ref Expression
elpw (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Proof of Theorem elpw
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elpw.1 . 2 𝐴 ∈ V
2 sseq1 3047 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 df-pw 3431 . 2 𝒫 𝐵 = {𝑥𝑥𝐵}
41, 2, 3elab2 2763 1 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wb 103  wcel 1438  Vcvv 2619  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:  selpw  3436  elpwg  3437  prsspw  3609  pwprss  3649  pwtpss  3650  pwv  3652  sspwuni  3813  iinpw  3819  iunpwss  3820  0elpw  3999  pwuni  4027  snelpw  4040  sspwb  4043  ssextss  4047  pwin  4109  pwunss  4110  iunpw  4302  xpsspw  4550  ssenen  6565  ioof  9387
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