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Theorem elpw2 4082
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
Hypothesis
Ref Expression
elpw2.1 𝐵 ∈ V
Assertion
Ref Expression
elpw2 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Proof of Theorem elpw2
StepHypRef Expression
1 elpw2.1 . 2 𝐵 ∈ V
2 elpw2g 4081 . 2 (𝐵 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
31, 2ax-mp 5 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  ax-sep 4046
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:  axpweq  4095  genpelxp  7326  ltexprlempr  7423  recexprlempr  7447  cauappcvgprlemcl  7468  cauappcvgprlemladd  7473  caucvgprlemcl  7491  caucvgprprlemcl  7519  uzf  9336  ixxf  9688  fzf  9801  cncfval  12738  reldvg  12827  dvfvalap  12829
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