Theorem elpwi2 5281

Description: Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof shortened by Wolf Lammen, 26-May-2024.)

Hypotheses

Ref	Expression
elpwi2.1	⊢ 𝐵 ∈ 𝑉
elpwi2.2	⊢ 𝐴 ⊆ 𝐵

Assertion

Ref	Expression
elpwi2	⊢ 𝐴 ∈ 𝒫 𝐵

Proof of Theorem elpwi2

Step	Hyp	Ref	Expression
1		elpwi2.2	. 2 ⊢ 𝐴 ⊆ 𝐵
2		elpwi2.1	. . . 4 ⊢ 𝐵 ∈ 𝑉
3	2	elexi 3464	. . 3 ⊢ 𝐵 ∈ V
4	3	elpw2 5280	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	mpbir 231	1 ⊢ 𝐴 ∈ 𝒫 𝐵

Syntax hints:   ∈ wcel 2114   ⊆ wss 3902  𝒫 cpw 4555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443  df-in 3909  df-ss 3919  df-pw 4557

This theorem is referenced by:  canth  7314  mptmpoopabbrd  8026  aceq3lem  10034  axdc3lem4  10367  uzf  12758  ixxf  13275  fzf  13431  bitsf  16358  prdsvallem  17378  prdsds  17388  wunnat  17887  ocvfval  21625  leordtval2  23160  cnpfval  23182  iscnp2  23187  islly2  23432  xkotf  23533  alexsubALTlem4  23998  sszcld  24766  bndth  24917  ishtpy  24931  fpwrelmap  32793  ballotlem2  34627  satfrnmapom  35545  cover2  37887  clsk1indlem1  44322  sprsymrelfolem1  47774