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Theorem elpw2 3938
 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 3937 . 2 (𝐵 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
31, 2ax-mp 7 1 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 102   ∈ wcel 1409  Vcvv 2574   ⊆ wss 2944  𝒫 cpw 3386 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388 This theorem is referenced by:  axpweq  3951  genpelxp  6666  ltexprlempr  6763  recexprlempr  6787  cauappcvgprlemcl  6808  cauappcvgprlemladd  6813  caucvgprlemcl  6831  caucvgprprlemcl  6859  uzf  8571  ixxf  8867  fzf  8979
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