Theorem wunxp

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

	Ref	Expression
Hypotheses	wun0.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	wunop.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
	wunop.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )
Assertion	wunxp	⊢ ( 𝜑 → ( 𝐴 × 𝐵 ) ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		wun0.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
2		wunop.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
3		wunop.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )
4	1 2 3	wunun	⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) ∈ 𝑈 )
5	1 4	wunpw	⊢ ( 𝜑 → 𝒫 ( 𝐴 ∪ 𝐵 ) ∈ 𝑈 )
6	1 5	wunpw	⊢ ( 𝜑 → 𝒫 𝒫 ( 𝐴 ∪ 𝐵 ) ∈ 𝑈 )
7		xpsspw	⊢ ( 𝐴 × 𝐵 ) ⊆ 𝒫 𝒫 ( 𝐴 ∪ 𝐵 )
8	7	a1i	⊢ ( 𝜑 → ( 𝐴 × 𝐵 ) ⊆ 𝒫 𝒫 ( 𝐴 ∪ 𝐵 ) )
9	1 6 8	wunss	⊢ ( 𝜑 → ( 𝐴 × 𝐵 ) ∈ 𝑈 )