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Metamath Proof Explorer

Theorem pwpwssunieq

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)

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

Proof

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