Theorem mnupwd 43016

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 5307	. . . . 5 ⊢ ∅ ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝜑 → ∅ ∈ V)
6	1, 2, 3, 5	mnuop23d 43015	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ 𝑈 (𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))))
7		simpl 483	. . . 4 ⊢ ((𝒫 𝐴 ⊆ 𝑤 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ 𝑈 (𝑖 ∈ 𝑣 ∧ 𝑣 ∈ ∅) → ∃𝑢 ∈ ∅ (𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤))) → 𝒫 𝐴 ⊆ 𝑤)
8	7	reximi 3084	. . 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 43013	1 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ⊆ wss 3948  ∅c0 4322  𝒫 cpw 4602  ∪ cuni 4908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323  df-pw 4604  df-uni 4909

This theorem is referenced by:  mnusnd  43017  mnuprssd  43018  mnugrud  43033