Theorem wunun 10750

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

Hypotheses

Ref	Expression
wununi.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wununi.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)
wunpr.3	⊢ (𝜑 → 𝐵 ∈ 𝑈)

Assertion

Ref	Expression
wunun	⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈)

Proof of Theorem wunun

Step	Hyp	Ref	Expression
1		wununi.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		wunpr.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
3		uniprg 4923	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
4	1, 2, 3	syl2anc 584	. 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
5		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
6	5, 1, 2	wunpr 10749	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
7	5, 6	wununi 10746	. 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ∈ 𝑈)
8	4, 7	eqeltrrd 2842	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   ∪ cun 3949  {cpr 4628  ∪ cuni 4907  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

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  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-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629  df-uni 4908  df-tr 5260  df-wun 10742

This theorem is referenced by:  wuntp  10751  wunsuc  10757  wunfi  10761  wunxp  10764  wuntpos  10774  wunsets  17214  catcoppccl  18162