Theorem elpwb 4567

Description: Characterization of the elements of a power class. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
elpwb	⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵))

Proof of Theorem elpwb

Step	Hyp	Ref	Expression
1		elex 3465	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ∈ V)
2		elpwg 4562	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
3	1, 2	biadanii 821	1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Vcvv 3444   ⊆ wss 3911  𝒫 cpw 4559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3446  df-ss 3928  df-pw 4561

This theorem is referenced by:  elpwpw  5061  elpwpwel  7723  onsupcl2  43207  onsupuni2  43212  onsupintrab2  43214  onuniintrab2  43217