Theorem elpwi2 5276

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 3452	. . 3 ⊢ 𝐵 ∈ V
4	3	elpw2 5275	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	mpbir 231	1 ⊢ 𝐴 ∈ 𝒫 𝐵

Syntax hints:   ∈ wcel 2114   ⊆ wss 3889  𝒫 cpw 4541

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 2708  ax-sep 5231

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 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-in 3896  df-ss 3906  df-pw 4543

This theorem is referenced by:  canth  7321  mptmpoopabbrd  8033  aceq3lem  10042  axdc3lem4  10375  uzf  12791  ixxf  13308  fzf  13465  bitsf  16396  prdsvallem  17417  prdsds  17427  wunnat  17926  ocvfval  21646  leordtval2  23177  cnpfval  23199  iscnp2  23204  islly2  23449  xkotf  23550  alexsubALTlem4  24015  sszcld  24783  bndth  24925  ishtpy  24939  fpwrelmap  32806  ballotlem2  34633  satfrnmapom  35552  cover2  38036  clsk1indlem1  44472  sprsymrelfolem1  47952