Theorem mnusnd 40127

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

Hypotheses

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

Assertion

Ref	Expression
mnusnd	⊢ (𝜑 → {𝐴} ∈ 𝑈)

Distinct variable groups:   𝑈,𝑘,𝑚,𝑛,𝑟,𝑝,𝑙   𝑈,𝑞,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 mnusnd

Step	Hyp	Ref	Expression
1		mnusnd.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		mnusnd.2	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
3		mnusnd.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	1, 2, 3	mnupwd 40126	. 2 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈)
5		snsspw 4686	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
6	5	a1i 11	. 2 ⊢ (𝜑 → {𝐴} ⊆ 𝒫 𝐴)
7	1, 2, 4, 6	mnussd 40122	1 ⊢ (𝜑 → {𝐴} ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1520   = wceq 1522   ∈ wcel 2081  {cab 2775  ∀wral 3105  ∃wrex 3106   ⊆ wss 3863  𝒫 cpw 4457  {csn 4476  ∪ cuni 4749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-v 3439  df-dif 3866  df-in 3870  df-ss 3878  df-nul 4216  df-pw 4459  df-sn 4477  df-uni 4750

This theorem is referenced by:  mnuprdlem4  40134