Theorem wunint 10696

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 485	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝑈 ∈ WUni)
3		wununi.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	1, 3	wununi 10687	. . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
5	4	adantr 485	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝑈)
6		intssuni 4936	. . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
7	6	adantl 486	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐴)
8	2, 5, 7	wunss 10693	1 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2149   ≠ wne 2964   ⊆ wss 3913  ∅c0 4294  ∪ cuni 4873  ∩ cint 4913  WUnicwun 10681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-in 3920  df-ss 3930  df-nul 4295  df-pw 4566  df-uni 4874  df-int 4914  df-tr 5220  df-wun 10683

This theorem is referenced by: (None)