Theorem elpwi2 5273

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 3453	. . 3 ⊢ 𝐵 ∈ V
4	3	elpw2 5272	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	mpbir 231	1 ⊢ 𝐴 ∈ 𝒫 𝐵

Syntax hints:   ∈ wcel 2114   ⊆ wss 3890  𝒫 cpw 4542

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 5232

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 3391  df-v 3432  df-in 3897  df-ss 3907  df-pw 4544

This theorem is referenced by:  canth  7315  mptmpoopabbrd  8027  aceq3lem  10036  axdc3lem4  10369  uzf  12785  ixxf  13302  fzf  13459  bitsf  16390  prdsvallem  17411  prdsds  17421  wunnat  17920  ocvfval  21659  leordtval2  23190  cnpfval  23212  iscnp2  23217  islly2  23462  xkotf  23563  alexsubALTlem4  24028  sszcld  24796  bndth  24938  ishtpy  24952  fpwrelmap  32824  ballotlem2  34652  satfrnmapom  35571  cover2  38053  clsk1indlem1  44493  sprsymrelfolem1  47967