Theorem rankxpl

Description: A lower bound on the rank of a Cartesian product. (Contributed by NM, 18-Sep-2006)

	Ref	Expression
Hypotheses	rankxpl.1	$$⊢A\in \mathrm{V}$$
	rankxpl.2	$$⊢B\in \mathrm{V}$$
Assertion	rankxpl	$$⊢A\times B\ne \varnothing \to \mathrm{rank}\left(A\cup B\right)\subseteq \mathrm{rank}\left(A\times B\right)$$

Proof

Step	Hyp	Ref	Expression
1		rankxpl.1	$$⊢A\in \mathrm{V}$$
2		rankxpl.2	$$⊢B\in \mathrm{V}$$
3		unixp	$$⊢A\times B\ne \varnothing \to \bigcup \bigcup \left(A\times B\right)=A\cup B$$
4	3	fveq2d	$$⊢A\times B\ne \varnothing \to \mathrm{rank}\left(\bigcup \bigcup \left(A\times B\right)\right)=\mathrm{rank}\left(A\cup B\right)$$
5	1 2	xpex	$$⊢A\times B\in \mathrm{V}$$
6	5	uniex	$$⊢\bigcup \left(A\times B\right)\in \mathrm{V}$$
7	6	rankuniss	$$⊢\mathrm{rank}\left(\bigcup \bigcup \left(A\times B\right)\right)\subseteq \mathrm{rank}\left(\bigcup \left(A\times B\right)\right)$$
8	5	rankuniss	$$⊢\mathrm{rank}\left(\bigcup \left(A\times B\right)\right)\subseteq \mathrm{rank}\left(A\times B\right)$$
9	7 8	sstri	$$⊢\mathrm{rank}\left(\bigcup \bigcup \left(A\times B\right)\right)\subseteq \mathrm{rank}\left(A\times B\right)$$
10	4 9	eqsstrrdi	$$⊢A\times B\ne \varnothing \to \mathrm{rank}\left(A\cup B\right)\subseteq \mathrm{rank}\left(A\times B\right)$$