Theorem wunun 10834

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 4888	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
4	1, 2, 3	syl2anc 596	. 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
5		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
6	5, 1, 2	wunpr 10833	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
7	5, 6	wununi 10830	. 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ∈ 𝑈)
8	4, 7	eqeltrrd 2865	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ∪ cun 4152  {cpr 4590  ∪ cuni 4872  WUnicwun 10824

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-v 3456  df-ss 3901  df-un 4154  df-sn 4589  df-pr 4591  df-uni 4873  df-tr 5219  df-wun 10826

This theorem is used by:  wuntp  10835  wunsuc  10841  wunfi  10845  wunxp  10848  wuntpos  10858  wunsets  17399  catcoppccl  18342