Theorem mnuop3d 44223

Description: Third operation of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypotheses

Ref	Expression
mnuop3d.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))))}
mnuop3d.2	⊢ (𝜑 → 𝑈 ∈ 𝑀)
mnuop3d.3	⊢ (𝜑 → 𝐴 ∈ 𝑈)
mnuop3d.4	⊢ (𝜑 → 𝐹 ⊆ 𝑈)

Assertion

Ref	Expression
mnuop3d	⊢ (𝜑 → ∃𝑤 ∈ 𝑈 ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))

Distinct variable groups:   𝑣,𝐹   𝑤,𝐴,𝑖   𝜑,𝑤,𝑣,𝑖   𝑤,𝑢,𝐹,𝑖   𝑤,𝑈,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙,𝑣,𝑖   𝑤,𝑟   𝑢,𝑈,𝑘,𝑚,𝑛,𝑟,𝑝,𝑙,𝑖

Allowed substitution hints:   𝜑(𝑢,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐴(𝑣,𝑢,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝐹(𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)   𝑀(𝑤,𝑣,𝑢,𝑖,𝑘,𝑚,𝑛,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem mnuop3d

Step	Hyp	Ref	Expression
1		mnuop3d.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))))}
2		mnuop3d.2	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
3		mnuop3d.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4		mnuop3d.4	. . . 4 ⊢ (𝜑 → 𝐹 ⊆ 𝑈)
5	2, 4	sselpwd 5308	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝒫 𝑈)
6	1, 2, 3, 5	mnuop23d 44218	. 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑈 (𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹) → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
7	4	sseld 3962	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ 𝐹 → 𝑣 ∈ 𝑈))
8	7	adantrd 491	. . . . . . . 8 ⊢ (𝜑 → ((𝑣 ∈ 𝐹 ∧ 𝑖 ∈ 𝑣) → 𝑣 ∈ 𝑈))
9		pm3.22 459	. . . . . . . 8 ⊢ ((𝑣 ∈ 𝐹 ∧ 𝑖 ∈ 𝑣) → (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹))
10	8, 9	jca2 513	. . . . . . 7 ⊢ (𝜑 → ((𝑣 ∈ 𝐹 ∧ 𝑖 ∈ 𝑣) → (𝑣 ∈ 𝑈 ∧ (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹))))
11	10	reximdv2 3151	. . . . . 6 ⊢ (𝜑 → (∃𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹)))
12	11	imim1d 82	. . . . 5 ⊢ (𝜑 → ((∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹) → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) → (∃𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
13	12	ralimdv 3156	. . . 4 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹) → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)) → ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
14	13	adantld 490	. . 3 ⊢ (𝜑 → ((𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹) → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) → ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
15	14	reximdv 3157	. 2 ⊢ (𝜑 → (∃𝑤 ∈ 𝑈 (𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹) → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) → ∃𝑤 ∈ 𝑈 ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
16	6, 15	mpd 15	1 ⊢ (𝜑 → ∃𝑤 ∈ 𝑈 ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝐹 𝑖 ∈ 𝑣 → ∃𝑢 ∈ 𝐹 (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2107  {cab 2712  ∀wral 3050  ∃wrex 3059   ⊆ wss 3931  𝒫 cpw 4580  ∪ cuni 4887

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 2706  ax-sep 5276

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 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-in 3938  df-ss 3948  df-pw 4582  df-uni 4888

This theorem is referenced by:  mnuprdlem4  44227  mnuunid  44229  mnurndlem2  44234