Theorem mnuunid 44301

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

Hypotheses

Ref	Expression
mnuunid.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))))}
mnuunid.2	⊢ (𝜑 → 𝑈 ∈ 𝑀)
mnuunid.3	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Assertion

Ref	Expression
mnuunid	⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)

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

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

Proof of Theorem mnuunid

Dummy variables 𝑣 𝑎 𝑤 𝑖 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mnuunid.1	. 2 ⊢ 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))))}
2		mnuunid.2	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
3		mnuunid.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	3	snssd 4808	. . . 4 ⊢ (𝜑 → {𝐴} ⊆ 𝑈)
5	1, 2, 3, 4	mnuop3d 44295	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ 𝑈 ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
6		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) → 𝑤 ∈ 𝑈)
7		sseq2 4009	. . . . 5 ⊢ (𝑎 = 𝑤 → (∪ 𝐴 ⊆ 𝑎 ↔ ∪ 𝐴 ⊆ 𝑤))
8	7	adantl 481	. . . 4 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ∧ 𝑎 = 𝑤) → (∪ 𝐴 ⊆ 𝑎 ↔ ∪ 𝐴 ⊆ 𝑤))
9		elssuni 4936	. . . . . . 7 ⊢ (𝑖 ∈ 𝐴 → 𝑖 ⊆ ∪ 𝐴)
10	9	rgen 3062	. . . . . 6 ⊢ ∀𝑖 ∈ 𝐴 𝑖 ⊆ ∪ 𝐴
11		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) → ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
12		eleq2 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝐴 → (𝑖 ∈ 𝑣 ↔ 𝑖 ∈ 𝐴))
13	12	rexsng 4675	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑈 → (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 ↔ 𝑖 ∈ 𝐴))
14	3, 13	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 ↔ 𝑖 ∈ 𝐴))
15		eleq2 2829	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝐴 → (𝑖 ∈ 𝑢 ↔ 𝑖 ∈ 𝐴))
16		unieq 4917	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝐴 → ∪ 𝑢 = ∪ 𝐴)
17	16	sseq1d 4014	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝐴 → (∪ 𝑢 ⊆ 𝑤 ↔ ∪ 𝐴 ⊆ 𝑤))
18	15, 17	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝐴 → ((𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ (𝑖 ∈ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑤)))
19	18	rexsng 4675	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝑈 → (∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ (𝑖 ∈ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑤)))
20	3, 19	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤) ↔ (𝑖 ∈ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑤)))
21	14, 20	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝜑 → ((∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ (𝑖 ∈ 𝐴 → (𝑖 ∈ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑤))))
22		anclb 545	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ 𝐴 → ∪ 𝐴 ⊆ 𝑤) ↔ (𝑖 ∈ 𝐴 → (𝑖 ∈ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑤)))
23	21, 22	bitr4di 289	. . . . . . . . . . 11 ⊢ (𝜑 → ((∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ (𝑖 ∈ 𝐴 → ∪ 𝐴 ⊆ 𝑤)))
24	23	imbi2d 340	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑖 ∈ 𝐴 → (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) ↔ (𝑖 ∈ 𝐴 → (𝑖 ∈ 𝐴 → ∪ 𝐴 ⊆ 𝑤))))
25		pm5.4 388	. . . . . . . . . 10 ⊢ ((𝑖 ∈ 𝐴 → (𝑖 ∈ 𝐴 → ∪ 𝐴 ⊆ 𝑤)) ↔ (𝑖 ∈ 𝐴 → ∪ 𝐴 ⊆ 𝑤))
26	24, 25	bitrdi 287	. . . . . . . . 9 ⊢ (𝜑 → ((𝑖 ∈ 𝐴 → (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) ↔ (𝑖 ∈ 𝐴 → ∪ 𝐴 ⊆ 𝑤)))
27	26	ralbidv2 3173	. . . . . . . 8 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ ∀𝑖 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑤))
28	27	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) → (∀𝑖 ∈ 𝐴 (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) ↔ ∀𝑖 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑤))
29	11, 28	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) → ∀𝑖 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑤)
30		sstr2 3989	. . . . . . 7 ⊢ (𝑖 ⊆ ∪ 𝐴 → (∪ 𝐴 ⊆ 𝑤 → 𝑖 ⊆ 𝑤))
31	30	ral2imi 3084	. . . . . 6 ⊢ (∀𝑖 ∈ 𝐴 𝑖 ⊆ ∪ 𝐴 → (∀𝑖 ∈ 𝐴 ∪ 𝐴 ⊆ 𝑤 → ∀𝑖 ∈ 𝐴 𝑖 ⊆ 𝑤))
32	10, 29, 31	mpsyl 68	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) → ∀𝑖 ∈ 𝐴 𝑖 ⊆ 𝑤)
33		unissb 4938	. . . . 5 ⊢ (∪ 𝐴 ⊆ 𝑤 ↔ ∀𝑖 ∈ 𝐴 𝑖 ⊆ 𝑤)
34	32, 33	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) → ∪ 𝐴 ⊆ 𝑤)
35	6, 8, 34	rspcedvd 3623	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ {𝐴}𝑖 ∈ 𝑣 → ∃𝑢 ∈ {𝐴} (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) → ∃𝑎 ∈ 𝑈 ∪ 𝐴 ⊆ 𝑎)
36	5, 35	rexlimddv 3160	. 2 ⊢ (𝜑 → ∃𝑎 ∈ 𝑈 ∪ 𝐴 ⊆ 𝑎)
37	1, 2, 36	mnuss2d 44288	1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2107  {cab 2713  ∀wral 3060  ∃wrex 3069   ⊆ wss 3950  𝒫 cpw 4599  {csn 4625  ∪ cuni 4906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-in 3957  df-ss 3967  df-pw 4601  df-sn 4626  df-uni 4907

This theorem is referenced by:  mnuund  44302  mnutrcld  44303  mnugrud  44308