Theorem mnupwd 44177

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

Hypotheses

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

Assertion

Ref	Expression
mnupwd	⊢ (𝜑 → 𝒫 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 mnupwd

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

Step	Hyp	Ref	Expression
1		mnupwd.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		mnupwd.2	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
3		mnupwd.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4		0ex 5328	. . . . 5 ⊢ ∅ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝜑 → ∅ ∈ V)
6	1, 2, 3, 5	mnuop23d 44176	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ 𝑈 (𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
7		simpl 482	. . . 4 ⊢ ((𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) → 𝒫 𝐴 ⊆ 𝑤)
8	7	reximi 3086	. . 3 ⊢ (∃𝑤 ∈ 𝑈 (𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) → ∃𝑤 ∈ 𝑈 𝒫 𝐴 ⊆ 𝑤)
9	6, 8	syl 17	. 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑈 𝒫 𝐴 ⊆ 𝑤)
10	1, 2, 9	mnuss2d 44174	1 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535   = wceq 1537   ∈ wcel 2103  {cab 2711  ∀wral 3063  ∃wrex 3072  Vcvv 3482   ⊆ wss 3970  ∅c0 4347  𝒫 cpw 4622  ∪ cuni 4931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-sep 5320  ax-nul 5327

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-in 3977  df-ss 3987  df-nul 4348  df-pw 4624  df-uni 4932

This theorem is referenced by:  mnusnd  44178  mnuprssd  44179  mnugrud  44194