Theorem wunint 10638

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

Hypotheses

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

Assertion

Ref	Expression
wunint	⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝑈)

Proof of Theorem wunint

Step	Hyp	Ref	Expression
1		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
2	1	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝑈 ∈ WUni)
3		wununi.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	1, 3	wununi 10629	. . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
5	4	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝑈)
6		intssuni 4927	. . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
7	6	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐴)
8	2, 5, 7	wunss 10635	1 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2114   ≠ wne 2933   ⊆ wss 3903  ∅c0 4287  ∪ cuni 4865  ∩ cint 4904  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 2709  ax-sep 5243

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-in 3910  df-ss 3920  df-nul 4288  df-pw 4558  df-uni 4866  df-int 4905  df-tr 5208  df-wun 10625

This theorem is referenced by: (None)