Theorem elpwbi 42854

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 5292	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
3	2	bicomi 226	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 208   ∈ wcel 2144  Vcvv 3456   ⊆ wss 3906  𝒫 cpw 4557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1101  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-in 3913  df-ss 3923  df-pw 4559

This theorem is referenced by: (None)