Theorem elpwbi 42333

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 5270	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
3	2	bicomi 224	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 206   ∈ wcel 2111  Vcvv 3436   ⊆ 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: (None)