Theorem unixpss

Description: The double class union of a Cartesian product is included in the union of its arguments. (Contributed by NM, 16-Sep-2006)

	Ref	Expression
Assertion	unixpss	⊢ ∪ ∪ ( 𝐴 × 𝐵 ) ⊆ ( 𝐴 ∪ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		xpsspw	⊢ ( 𝐴 × 𝐵 ) ⊆ 𝒫 𝒫 ( 𝐴 ∪ 𝐵 )
2	1	unissi	⊢ ∪ ( 𝐴 × 𝐵 ) ⊆ ∪ 𝒫 𝒫 ( 𝐴 ∪ 𝐵 )
3		unipw	⊢ ∪ 𝒫 𝒫 ( 𝐴 ∪ 𝐵 ) = 𝒫 ( 𝐴 ∪ 𝐵 )
4	2 3	sseqtri	⊢ ∪ ( 𝐴 × 𝐵 ) ⊆ 𝒫 ( 𝐴 ∪ 𝐵 )
5	4	unissi	⊢ ∪ ∪ ( 𝐴 × 𝐵 ) ⊆ ∪ 𝒫 ( 𝐴 ∪ 𝐵 )
6		unipw	⊢ ∪ 𝒫 ( 𝐴 ∪ 𝐵 ) = ( 𝐴 ∪ 𝐵 )
7	5 6	sseqtri	⊢ ∪ ∪ ( 𝐴 × 𝐵 ) ⊆ ( 𝐴 ∪ 𝐵 )