Theorem wunin 10611

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 4186	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐴)
5	1, 2, 4	wunss 10610	1 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2113   ∩ cin 3897   ⊆ wss 3898  WUnicwun 10598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-in 3905  df-ss 3915  df-pw 4551  df-uni 4859  df-tr 5201  df-wun 10600

This theorem is referenced by:  wunress  17162