Theorem elpwpw 5059

Description: Characterization of the elements of a double power class: they are exactly the sets whose union is included in that class. (Contributed by BJ, 29-Apr-2021.)

Assertion

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

Proof of Theorem elpwpw

Step	Hyp	Ref	Expression
1		elpwb 4563	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵))
2		sspwuni 5057	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
3	2	anbi2i 632	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵) ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
4	1, 3	bitri 277	1 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 399   ∈ wcel 2142  Vcvv 3454   ⊆ wss 3904  𝒫 cpw 4555  ∪ cuni 4865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-v 3456  df-ss 3921  df-pw 4557  df-uni 4866

This theorem is referenced by:  pwpwab  5060  elpwpwel  7750  ismnushort  44877