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Metamath Proof Explorer

Theorem pwunss

Description: The power class of the union of two classes includes the union of their power classes. Exercise 4.12(k) of Mendelson p. 235. (Contributed by NM, 23-Nov-2003) Remove use of ax-sep , ax-nul , ax-pr and shorten proof. (Revised by BJ, 13-Apr-2024)

		Ref	Expression
	Assertion	pwunss	$⊢ 𝒫 A \cup 𝒫 B \subseteq 𝒫 (A \cup B)$

Proof

Step	Hyp	Ref	Expression
1		ssun1	$⊢ A \subseteq A \cup B$
2	1	sspwi	$⊢ 𝒫 A \subseteq 𝒫 (A \cup B)$
3		ssun2	$⊢ B \subseteq A \cup B$
4	3	sspwi	$⊢ 𝒫 B \subseteq 𝒫 (A \cup B)$
5	2 4	unssi	$⊢ 𝒫 A \cup 𝒫 B \subseteq 𝒫 (A \cup B)$