Theorem wunint 9537

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 481	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝑈 ∈ WUni)
3		wununi.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	1, 3	wununi 9528	. . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)
5	4	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝑈)
6		intssuni 4499	. . 3 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
7	6	adantl 482	. 2 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐴)
8	2, 5, 7	wunss 9534	1 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990   ≠ wne 2794   ⊆ wss 3574  ∅c0 3915  ∪ cuni 4436  ∩ cint 4475  WUnicwun 9522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-uni 4437  df-int 4476  df-tr 4753  df-wun 9524

This theorem is referenced by: (None)