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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 ∪ 𝒫 𝐵 ) ⊆ 𝒫 ( 𝐴 ∪ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ssun1	⊢ 𝐴 ⊆ ( 𝐴 ∪ 𝐵 )
2	1	sspwi	⊢ 𝒫 𝐴 ⊆ 𝒫 ( 𝐴 ∪ 𝐵 )
3		ssun2	⊢ 𝐵 ⊆ ( 𝐴 ∪ 𝐵 )
4	3	sspwi	⊢ 𝒫 𝐵 ⊆ 𝒫 ( 𝐴 ∪ 𝐵 )
5	2 4	unssi	⊢ ( 𝒫 𝐴 ∪ 𝒫 𝐵 ) ⊆ 𝒫 ( 𝐴 ∪ 𝐵 )