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Theorem elpw 3596
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 3193 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 df-pw 3592 . 2 𝒫 𝐵 = {𝑥𝑥𝐵}
41, 2, 3elab2 2900 1 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wb 105  wcel 2160  Vcvv 2752  wss 3144  𝒫 cpw 3590
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592
This theorem is referenced by:  velpw  3597  elpwg  3598  prsspw  3780  pwprss  3820  pwtpss  3821  pwv  3823  sspwuni  3986  iinpw  3992  iunpwss  3993  0elpw  4182  pwuni  4210  snelpw  4231  sspwb  4234  ssextss  4238  pwin  4300  pwunss  4301  iunpw  4498  xpsspw  4756  ssenen  6879  pw1ne3  7259  3nsssucpw1  7265  ioof  10001  tgdom  14029  distop  14042  epttop  14047  resttopon  14128  txuni2  14213
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