Theorem elpwbi 43087

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 5303	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)
3	2	bicomi 227	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902  𝒫 cpw 4560

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-in 3909  df-ss 3919  df-pw 4562

This theorem is used by: (None)