Theorem pwpwssunieq 5002

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

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem pwpwssunieq

Step	Hyp	Ref	Expression
1		eqimss 3947	. . 3 ⊢ (∪ 𝑥 = 𝐴 → ∪ 𝑥 ⊆ 𝐴)
2	1	ss2abi 3970	. 2 ⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴}
3		pwpwab 5001	. 2 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴}
4	2, 3	sseqtrri 3928	1 ⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴

Syntax hints:   = wceq 1543  {cab 2712   ⊆ wss 3857  𝒫 cpw 4503  ∪ cuni 4809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2706

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-ral 3059  df-v 3403  df-in 3864  df-ss 3874  df-pw 4505  df-uni 4810

This theorem is referenced by:  toponsspwpw  21791  dmtopon  21792