Theorem wunint 10755

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 10746	. . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
5	4	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝑈)
6		intssuni 4970	. . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
7	6	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐴)
8	2, 5, 7	wunss 10752	1 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108   ≠ wne 2940   ⊆ wss 3951  ∅c0 4333  ∪ cuni 4907  ∩ cint 4946  WUnicwun 10740

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-pw 4602  df-uni 4908  df-int 4947  df-tr 5260  df-wun 10742

This theorem is referenced by: (None)