Theorem pwpwssunieq 5107

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 4040	. . 3 ⊢ (∪ 𝑥 = 𝐴 → ∪ 𝑥 ⊆ 𝐴)
2	1	ss2abi 4063	. 2 ⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴}
3		pwpwab 5106	. 2 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴}
4	2, 3	sseqtrri 4019	1 ⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴

Syntax hints:   = wceq 1542  {cab 2710   ⊆ wss 3948  𝒫 cpw 4602  ∪ cuni 4908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-in 3955  df-ss 3965  df-pw 4604  df-uni 4909

This theorem is referenced by:  toponsspwpw  22416  dmtopon  22417