Theorem elpwb 4570

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 3476	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ∈ V)
2		elpwg 4565	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
3	1, 2	biadanii 833	1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3922  df-pw 4564

This theorem is referenced by:  elpwpw  5068  elpwpwel  7762  onsupcl2  43972  onsupuni2  43977  onsupintrab2  43979  onuniintrab2  43982