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Theorem mnuunid 40123
Description: Minimal universes are closed under union. (Contributed by Rohan Ridenour, 13-Aug-2023.)
Hypotheses
Ref Expression
mnuunid.1 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
mnuunid.2 (𝜑𝑈𝑀)
mnuunid.3 (𝜑𝐴𝑈)
Assertion
Ref Expression
mnuunid (𝜑 𝐴𝑈)
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.
StepHypRef Expression
1 mnuunid.1 . 2 𝑀 = {𝑘 ∣ ∀𝑙𝑘 (𝒫 𝑙𝑘 ∧ ∀𝑚𝑛𝑘 (𝒫 𝑙𝑛 ∧ ∀𝑝𝑙 (∃𝑞𝑘 (𝑝𝑞𝑞𝑚) → ∃𝑟𝑚 (𝑝𝑟 𝑟𝑛))))}
2 mnuunid.2 . 2 (𝜑𝑈𝑀)
3 mnuunid.3 . . . 4 (𝜑𝐴𝑈)
43snssd 4651 . . . 4 (𝜑 → {𝐴} ⊆ 𝑈)
51, 2, 3, 4mnuop3d 40117 . . 3 (𝜑 → ∃𝑤𝑈𝑖𝐴 (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)))
6 simprl 767 . . . 4 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)))) → 𝑤𝑈)
7 sseq2 3916 . . . . 5 (𝑎 = 𝑤 → ( 𝐴𝑎 𝐴𝑤))
87adantl 482 . . . 4 (((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)))) ∧ 𝑎 = 𝑤) → ( 𝐴𝑎 𝐴𝑤))
9 elssuni 4776 . . . . . . 7 (𝑖𝐴𝑖 𝐴)
109rgen 3114 . . . . . 6 𝑖𝐴 𝑖 𝐴
11 simprr 769 . . . . . . 7 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)))) → ∀𝑖𝐴 (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)))
12 eleq2 2870 . . . . . . . . . . . . . . 15 (𝑣 = 𝐴 → (𝑖𝑣𝑖𝐴))
1312rexsng 4523 . . . . . . . . . . . . . 14 (𝐴𝑈 → (∃𝑣 ∈ {𝐴}𝑖𝑣𝑖𝐴))
143, 13syl 17 . . . . . . . . . . . . 13 (𝜑 → (∃𝑣 ∈ {𝐴}𝑖𝑣𝑖𝐴))
15 eleq2 2870 . . . . . . . . . . . . . . . 16 (𝑢 = 𝐴 → (𝑖𝑢𝑖𝐴))
16 unieq 4755 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝐴 𝑢 = 𝐴)
1716sseq1d 3921 . . . . . . . . . . . . . . . 16 (𝑢 = 𝐴 → ( 𝑢𝑤 𝐴𝑤))
1815, 17anbi12d 630 . . . . . . . . . . . . . . 15 (𝑢 = 𝐴 → ((𝑖𝑢 𝑢𝑤) ↔ (𝑖𝐴 𝐴𝑤)))
1918rexsng 4523 . . . . . . . . . . . . . 14 (𝐴𝑈 → (∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤) ↔ (𝑖𝐴 𝐴𝑤)))
203, 19syl 17 . . . . . . . . . . . . 13 (𝜑 → (∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤) ↔ (𝑖𝐴 𝐴𝑤)))
2114, 20imbi12d 346 . . . . . . . . . . . 12 (𝜑 → ((∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)) ↔ (𝑖𝐴 → (𝑖𝐴 𝐴𝑤))))
22 anclb 546 . . . . . . . . . . . 12 ((𝑖𝐴 𝐴𝑤) ↔ (𝑖𝐴 → (𝑖𝐴 𝐴𝑤)))
2321, 22syl6bbr 290 . . . . . . . . . . 11 (𝜑 → ((∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)) ↔ (𝑖𝐴 𝐴𝑤)))
2423imbi2d 342 . . . . . . . . . 10 (𝜑 → ((𝑖𝐴 → (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤))) ↔ (𝑖𝐴 → (𝑖𝐴 𝐴𝑤))))
25 pm5.4 390 . . . . . . . . . 10 ((𝑖𝐴 → (𝑖𝐴 𝐴𝑤)) ↔ (𝑖𝐴 𝐴𝑤))
2624, 25syl6bb 288 . . . . . . . . 9 (𝜑 → ((𝑖𝐴 → (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤))) ↔ (𝑖𝐴 𝐴𝑤)))
2726ralbidv2 3161 . . . . . . . 8 (𝜑 → (∀𝑖𝐴 (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)) ↔ ∀𝑖𝐴 𝐴𝑤))
2827adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)))) → (∀𝑖𝐴 (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)) ↔ ∀𝑖𝐴 𝐴𝑤))
2911, 28mpbid 233 . . . . . 6 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)))) → ∀𝑖𝐴 𝐴𝑤)
30 sstr2 3898 . . . . . . 7 (𝑖 𝐴 → ( 𝐴𝑤𝑖𝑤))
3130ral2imi 3122 . . . . . 6 (∀𝑖𝐴 𝑖 𝐴 → (∀𝑖𝐴 𝐴𝑤 → ∀𝑖𝐴 𝑖𝑤))
3210, 29, 31mpsyl 68 . . . . 5 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)))) → ∀𝑖𝐴 𝑖𝑤)
33 unissb 4778 . . . . 5 ( 𝐴𝑤 ↔ ∀𝑖𝐴 𝑖𝑤)
3432, 33sylibr 235 . . . 4 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)))) → 𝐴𝑤)
356, 8, 34rspcedvd 3564 . . 3 ((𝜑 ∧ (𝑤𝑈 ∧ ∀𝑖𝐴 (∃𝑣 ∈ {𝐴}𝑖𝑣 → ∃𝑢 ∈ {𝐴} (𝑖𝑢 𝑢𝑤)))) → ∃𝑎𝑈 𝐴𝑎)
365, 35rexlimddv 3253 . 2 (𝜑 → ∃𝑎𝑈 𝐴𝑎)
371, 2, 36mnuss2d 40110 1 (𝜑 𝐴𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1520   = wceq 1522  wcel 2080  {cab 2774  wral 3104  wrex 3105  wss 3861  𝒫 cpw 4455  {csn 4474   cuni 4747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-v 3438  df-sbc 3708  df-in 3868  df-ss 3876  df-pw 4457  df-sn 4475  df-uni 4748
This theorem is referenced by:  mnuund  40124  mnutrcld  40125  mnugrud  40130
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