Theorem elpwi2 5271

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 3459	. . 3 ⊢ 𝐵 ∈ V
4	3	elpw2 5270	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	mpbir 231	1 ⊢ 𝐴 ∈ 𝒫 𝐵

Syntax hints:   ∈ wcel 2111   ⊆ wss 3897  𝒫 cpw 4547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-in 3904  df-ss 3914  df-pw 4549

This theorem is referenced by:  canth  7300  mptmpoopabbrd  8012  aceq3lem  10011  axdc3lem4  10344  uzf  12735  ixxf  13255  fzf  13411  bitsf  16338  prdsvallem  17358  prdsds  17368  wunnat  17866  ocvfval  21603  leordtval2  23127  cnpfval  23149  iscnp2  23154  islly2  23399  xkotf  23500  alexsubALTlem4  23965  sszcld  24733  bndth  24884  ishtpy  24898  fpwrelmap  32716  ballotlem2  34502  satfrnmapom  35414  cover2  37765  clsk1indlem1  44148  sprsymrelfolem1  47602