Theorem elpwb 4555

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 3457	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ∈ V)
2		elpwg 4550	. 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 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

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-ss 3914  df-pw 4549

This theorem is referenced by:  elpwpw  5048  elpwpwel  7700  onsupcl2  43328  onsupuni2  43333  onsupintrab2  43335  onuniintrab2  43338