Theorem mnusnd 45092

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 45091	. 2 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈)
5		snsspw 4804	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
6	5	a1i 11	. 2 ⊢ (𝜑 → {𝐴} ⊆ 𝒫 𝐴)
7	1, 2, 4, 6	mnussd 45087	1 ⊢ (𝜑 → {𝐴} ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899  𝒫 cpw 4557  {csn 4584  ∪ cuni 4867

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-nul 5263

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-in 3906  df-ss 3916  df-nul 4280  df-pw 4559  df-sn 4585  df-uni 4868

This theorem is used by:  mnuprdlem4  45099