Theorem elpwi2 4253

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 2816	. . 3 ⊢ 𝐵 ∈ V
4	3	elpw2 4252	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	mpbir 146	1 ⊢ 𝐴 ∈ 𝒫 𝐵

Syntax hints:   ∈ wcel 2202   ⊆ wss 3201  𝒫 cpw 3656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-sep 4212

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-pw 3658

This theorem is referenced by:  canth  5979  bitsf  12570  prdsvallem  13418