Theorem wunin 10732

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

Hypotheses

Ref	Expression
wununi.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wununi.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Assertion

Ref	Expression
wunin	⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑈)

Proof of Theorem wunin

Step	Hyp	Ref	Expression
1		wununi.1	. 2 ⊢ (𝜑 → 𝑈 ∈ WUni)
2		wununi.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
3		inss1 4217	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐴)
5	1, 2, 4	wunss 10731	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2109   ∩ cin 3930   ⊆ wss 3931  WUnicwun 10719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-in 3938  df-ss 3948  df-pw 4582  df-uni 4889  df-tr 5235  df-wun 10721

This theorem is referenced by:  wunress  17275