Theorem pwpwab 3846

Description: The double power class written as a class abstraction: the class of sets whose union is included in the given class. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
pwpwab	⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴}

Distinct variable group:   𝑥,𝐴

Proof of Theorem pwpwab

Step	Hyp	Ref	Expression
1		vex 2644	. . 3 ⊢ 𝑥 ∈ V
2		elpwpw 3845	. . 3 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ (𝑥 ∈ V ∧ ∪ 𝑥 ⊆ 𝐴))
3	1, 2	mpbiran 892	. 2 ⊢ (𝑥 ∈ 𝒫 𝒫 𝐴 ↔ ∪ 𝑥 ⊆ 𝐴)
4	3	abbi2i 2214	1 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴}

Syntax hints:   = wceq 1299   ∈ wcel 1448  {cab 2086  Vcvv 2641   ⊆ wss 3021  𝒫 cpw 3457  ∪ cuni 3683

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459  df-uni 3684

This theorem is referenced by:  pwpwssunieq  3847