Theorem wunxp

Description: A weak universe is closed under cartesian products. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	wun0.1	$$⊢\phi \to U\in \mathrm{WUni}$$
	wunop.2	$$⊢\phi \to A\in U$$
	wunop.3	$$⊢\phi \to B\in U$$
Assertion	wunxp	$$⊢\phi \to A\times B\in U$$

Proof

Step	Hyp	Ref	Expression
1		wun0.1	$$⊢\phi \to U\in \mathrm{WUni}$$
2		wunop.2	$$⊢\phi \to A\in U$$
3		wunop.3	$$⊢\phi \to B\in U$$
4	1 2 3	wunun	$$⊢\phi \to A\cup B\in U$$
5	1 4	wunpw	$$⊢\phi \to 𝒫\left(A\cup B\right)\in U$$
6	1 5	wunpw	$$⊢\phi \to 𝒫𝒫\left(A\cup B\right)\in U$$
7		xpsspw	$$⊢A\times B\subseteq 𝒫𝒫\left(A\cup B\right)$$
8	7	a1i	$$⊢\phi \to A\times B\subseteq 𝒫𝒫\left(A\cup B\right)$$
9	1 6 8	wunss	$$⊢\phi \to A\times B\in U$$