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Theorem elpw 3516
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 3120 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 df-pw 3512 . 2 𝒫 𝐵 = {𝑥𝑥𝐵}
41, 2, 3elab2 2832 1 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 1480  Vcvv 2686  wss 3071  𝒫 cpw 3510
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512
This theorem is referenced by:  velpw  3517  elpwg  3518  prsspw  3692  pwprss  3732  pwtpss  3733  pwv  3735  sspwuni  3897  iinpw  3903  iunpwss  3904  0elpw  4088  pwuni  4116  snelpw  4135  sspwb  4138  ssextss  4142  pwin  4204  pwunss  4205  iunpw  4401  xpsspw  4651  ssenen  6745  ioof  9766  tgdom  12255  distop  12268  epttop  12273  resttopon  12354  txuni2  12439
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