Theorem elpwbi 42730

Description: Membership in a power set, biconditional. (Contributed by Steven Nguyen, 17-Jul-2022.) (Proof shortened by Steven Nguyen, 16-Sep-2022.)

Hypothesis

Ref	Expression
elpwbi.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
elpwbi	⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwbi

Step	Hyp	Ref	Expression
1		elpwbi.1	. . 3 ⊢ 𝐵 ∈ V
2	1	elpw2 5264	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
3	2	bicomi 226	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 208   ∈ wcel 2121  Vcvv 3433   ⊆ wss 3884  𝒫 cpw 4531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220

This theorem depends on definitions:  df-bi 209  df-an 398  df-3an 1095  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-v 3435  df-in 3891  df-ss 3901  df-pw 4533

This theorem is referenced by: (None)