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	$$⊢\bigcup \bigcup \left(A\times B\right)\subseteq A\cup B$$

Proof

Step	Hyp	Ref	Expression
1		xpsspw	$$⊢A\times B\subseteq 𝒫𝒫\left(A\cup B\right)$$
2	1	unissi	$$⊢\bigcup \left(A\times B\right)\subseteq \bigcup 𝒫𝒫\left(A\cup B\right)$$
3		unipw	$$⊢\bigcup 𝒫𝒫\left(A\cup B\right)=𝒫\left(A\cup B\right)$$
4	2 3	sseqtri	$$⊢\bigcup \left(A\times B\right)\subseteq 𝒫\left(A\cup B\right)$$
5	4	unissi	$$⊢\bigcup \bigcup \left(A\times B\right)\subseteq \bigcup 𝒫\left(A\cup B\right)$$
6		unipw	$$⊢\bigcup 𝒫\left(A\cup B\right)=A\cup B$$
7	5 6	sseqtri	$$⊢\bigcup \bigcup \left(A\times B\right)\subseteq A\cup B$$