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Theorem elpwg 3595
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 2250 . 2 (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵𝐴 ∈ 𝒫 𝐵))
2 sseq1 3190 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 vex 2752 . . 3 𝑥 ∈ V
43elpw 3593 . 2 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
51, 2, 4vtoclbg 2810 1 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wcel 2158  wss 3141  𝒫 cpw 3587
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-in 3147  df-ss 3154  df-pw 3589
This theorem is referenced by:  elpwi  3596  elpwb  3597  pwidg  3601  prsspwg  3764  elpw2g  4168  snelpwi  4224  prelpwi  4226  pwel  4230  eldifpw  4489  f1opw2  6091  2pwuninelg  6298  tfrlemibfn  6343  tfr1onlembfn  6359  tfrcllembfn  6372  elpmg  6678  fopwdom  6850  fiinopn  13800  ssntr  13918
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