Theorem mnupwd 45017

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 5275	. . . . 5 ⊢ ∅ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝜑 → ∅ ∈ V)
6	1, 2, 3, 5	mnuop23d 45016	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ 𝑈 (𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
7		simpl 488	. . . 4 ⊢ ((𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) → 𝒫 𝐴 ⊆ 𝑤)
8	7	reximi 3106	. . 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 18	. 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑈 𝒫 𝐴 ⊆ 𝑤)
10	1, 2, 9	mnuss2d 45014	1 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146  {cab 2744  ∀wral 3082  ∃wrex 3092  Vcvv 3458   ⊆ wss 3908  ∅c0 4289  𝒫 cpw 4567  ∪ cuni 4877

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-in 3915  df-ss 3925  df-nul 4290  df-pw 4569  df-uni 4878

This theorem is used by:  mnusnd  45018  mnuprssd  45019  mnugrud  45034