Theorem wunin

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

	Ref	Expression
Hypotheses	wununi.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	wununi.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
Assertion	wunin	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		wununi.1	⊢ ( 𝜑 → 𝑈 ∈ WUni )
2		wununi.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
3		inss1	⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴
4	3	a1i	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 )
5	1 2 4	wunss	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) ∈ 𝑈 )