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Theorem elpwg 3598
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 2252 . 2 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵𝐴 ∈ 𝒫 𝐵))
2 sseq1 3193 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 vex 2755 . . 3 𝑥 ∈ V
43elpw 3596 . 2 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
51, 2, 4vtoclbg 2813 1 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wcel 2160  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:  elpwi  3599  elpwb  3600  pwidg  3604  prsspwg  3767  elpw2g  4174  snelpwi  4230  prelpwi  4232  pwel  4236  eldifpw  4495  f1opw2  6100  2pwuninelg  6308  tfrlemibfn  6353  tfr1onlembfn  6369  tfrcllembfn  6382  elpmg  6690  pw2f1odclem  6862  fopwdom  6864  fiinopn  13964  ssntr  14082
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