Theorem elpwb 3636

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 2788	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ∈ V)
2		elpwg 3634	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
3	1, 2	biadan2 456	1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∈ wcel 2178  Vcvv 2776   ⊆ wss 3174  𝒫 cpw 3626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180  df-ss 3187  df-pw 3628

This theorem is referenced by:  elpwpw  4028  elpwpwel  4540