Theorem elpwpw 3973

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 3585	. 2 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵))
2		sspwuni 3971	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
3	2	anbi2i 457	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐴 ⊆ 𝒫 𝐵) ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))
4	1, 3	bitri 184	1 ⊢ (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ (𝐴 ∈ V ∧ ∪ 𝐴 ⊆ 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∈ wcel 2148  Vcvv 2737   ⊆ wss 3129  𝒫 cpw 3575  ∪ cuni 3809

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-in 3135  df-ss 3142  df-pw 3577  df-uni 3810

This theorem is referenced by:  pwpwab  3974  elpwpwel  4475