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Theorem elpw 3581
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 3178 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 df-pw 3577 . 2 𝒫 𝐵 = {𝑥𝑥𝐵}
41, 2, 3elab2 2885 1 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wb 105  wcel 2148  Vcvv 2737  wss 3129  𝒫 cpw 3575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-pw 3577
This theorem is referenced by:  velpw  3582  elpwg  3583  prsspw  3765  pwprss  3805  pwtpss  3806  pwv  3808  sspwuni  3970  iinpw  3976  iunpwss  3977  0elpw  4163  pwuni  4191  snelpw  4212  sspwb  4215  ssextss  4219  pwin  4281  pwunss  4282  iunpw  4479  xpsspw  4737  ssenen  6847  pw1ne3  7225  3nsssucpw1  7231  ioof  9966  tgdom  13434  distop  13447  epttop  13452  resttopon  13533  txuni2  13618
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