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Metamath Proof Explorer

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 )