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Theorem pwel 4045
Description: Membership of a power class. Exercise 10 of [Enderton] p. 26. (Contributed by NM, 13-Jan-2007.)
Assertion
Ref Expression
pwel (𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)

Proof of Theorem pwel
StepHypRef Expression
1 elssuni 3681 . . 3 (𝐴𝐵𝐴 𝐵)
2 sspwb 4043 . . 3 (𝐴 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
31, 2sylib 120 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
4 pwexg 4015 . . 3 (𝐴𝐵 → 𝒫 𝐴 ∈ V)
5 elpwg 3437 . . 3 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵))
64, 5syl 14 . 2 (𝐴𝐵 → (𝒫 𝐴 ∈ 𝒫 𝒫 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵))
73, 6mpbird 165 1 (𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wcel 1438  Vcvv 2619  wss 2999  𝒫 cpw 3429   cuni 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-uni 3654
This theorem is referenced by: (None)
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