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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