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Theorem elpwd 38264
 Description: Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
elpwd.1 (𝜑𝐴𝑉)
elpwd.2 (𝜑𝐴𝐵)
Assertion
Ref Expression
elpwd (𝜑𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwd
StepHypRef Expression
1 elpwd.2 . 2 (𝜑𝐴𝐵)
2 elpwd.1 . . 3 (𝜑𝐴𝑉)
3 elpwg 4116 . . 3 (𝐴𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
42, 3syl 17 . 2 (𝜑 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
51, 4mpbird 246 1 (𝜑𝐴 ∈ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977   ⊆ wss 3540  𝒫 cpw 4108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110 This theorem is referenced by:  dmvolss  38878  sge0xaddlem1  39326  ovnval2b  39442  ovnsubadd2lem  39535  vonvolmbllem  39550  vonvolmbl  39551  smfresal  39673  smfpimbor1lem1  39683
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