Theorem elpwbi 42223

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 5352	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
3	2	bicomi 224	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 206   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-pw 4624

This theorem is referenced by: (None)