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Theorem pwel 4173
 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 3796 . . 3 (𝐴𝐵𝐴 𝐵)
2 sspwb 4171 . . 3 (𝐴 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
31, 2sylib 121 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
4 pwexg 4136 . . 3 (𝐴𝐵 → 𝒫 𝐴 ∈ V)
5 elpwg 3547 . . 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 166 1 (𝐴𝐵 → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∈ wcel 2125  Vcvv 2709   ⊆ wss 3098  𝒫 cpw 3539  ∪ cuni 3768 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-uni 3769 This theorem is referenced by: (None)
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