Theorem wunun 10601

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 4875	. . 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 10600	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
7	5, 6	wununi 10597	. 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ∈ 𝑈)
8	4, 7	eqeltrrd 2832	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ∪ cun 3900  {cpr 4578  ∪ cuni 4859  WUnicwun 10591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-v 3438  df-un 3907  df-ss 3919  df-sn 4577  df-pr 4579  df-uni 4860  df-tr 5199  df-wun 10593

This theorem is referenced by:  wuntp  10602  wunsuc  10608  wunfi  10612  wunxp  10615  wuntpos  10625  wunsets  17088  catcoppccl  18024