Theorem elpwi2 5335

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 3503	. . 3 ⊢ 𝐵 ∈ V
4	3	elpw2 5334	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	mpbir 231	1 ⊢ 𝐴 ∈ 𝒫 𝐵

Syntax hints:   ∈ wcel 2108   ⊆ wss 3951  𝒫 cpw 4600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pw 4602

This theorem is referenced by:  canth  7385  mptmpoopabbrd  8105  aceq3lem  10160  axdc3lem4  10493  uzf  12881  ixxf  13397  fzf  13551  bitsf  16464  prdsvallem  17499  prdsds  17509  wunnat  18004  ocvfval  21684  leordtval2  23220  cnpfval  23242  iscnp2  23247  islly2  23492  xkotf  23593  alexsubALTlem4  24058  sszcld  24839  bndth  24990  ishtpy  25004  fpwrelmap  32744  ballotlem2  34491  satfrnmapom  35375  cover2  37722  clsk1indlem1  44058  sprsymrelfolem1  47479