Theorem mnuund 43339

Description: Minimal universes are closed under binary unions. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypotheses

Ref	Expression
mnuund.1	⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
mnuund.2	⊢ (𝜑 → 𝑈 ∈ 𝑀)
mnuund.3	⊢ (𝜑 → 𝐴 ∈ 𝑈)
mnuund.4	⊢ (𝜑 → 𝐵 ∈ 𝑈)

Assertion

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

Distinct variable groups:   𝑈,𝑘,𝑚,𝑛,𝑟,𝑝,𝑙   𝑈,𝑞,𝑘,𝑚,𝑛,𝑝,𝑙

Allowed substitution hints:   𝜑(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐴(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐵(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑀(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnuund

Step	Hyp	Ref	Expression
1		mnuund.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		mnuund.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
3		uniprg 4925	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
4	1, 2, 3	syl2anc 584	. 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
5		mnuund.1	. . 3 ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
6		mnuund.2	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
7	5, 6, 1, 2	mnuprd 43337	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
8	5, 6, 7	mnuunid 43338	. 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ∈ 𝑈)
9	4, 8	eqeltrrd 2834	1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070   ∪ cun 3946   ⊆ wss 3948  𝒫 cpw 4602  {cpr 4630  ∪ cuni 4908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-pw 4604  df-sn 4629  df-pr 4631  df-uni 4909

This theorem is referenced by: (None)