Theorem wunin 10636

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 4177	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐴)
5	1, 2, 4	wunss 10635	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2114   ∩ cin 3888   ⊆ wss 3889  WUnicwun 10623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-in 3896  df-ss 3906  df-pw 4543  df-uni 4851  df-tr 5193  df-wun 10625

This theorem is referenced by:  wunress  17219