Theorem wunun 10670

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 4890	. . 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 10669	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
7	5, 6	wununi 10666	. 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ∈ 𝑈)
8	4, 7	eqeltrrd 2830	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∪ cun 3915  {cpr 4594  ∪ cuni 4874  WUnicwun 10660

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-v 3452  df-un 3922  df-ss 3934  df-sn 4593  df-pr 4595  df-uni 4875  df-tr 5218  df-wun 10662

This theorem is referenced by:  wuntp  10671  wunsuc  10677  wunfi  10681  wunxp  10684  wuntpos  10694  wunsets  17154  catcoppccl  18086