Theorem mnusnd 44808

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 44807	. 2 ⊢ (𝜑 → 𝒫 𝐴 ∈ 𝑈)
5		snsspw 4801	. . 3 ⊢ {𝐴} ⊆ 𝒫 𝐴
6	5	a1i 11	. 2 ⊢ (𝜑 → {𝐴} ⊆ 𝒫 𝐴)
7	1, 2, 4, 6	mnussd 44803	1 ⊢ (𝜑 → {𝐴} ∈ 𝑈)

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1557   = wceq 1559   ∈ wcel 2141  {cab 2739  ∀wral 3075  ∃wrex 3085   ⊆ wss 3904  𝒫 cpw 4554  {csn 4581  ∪ cuni 4864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-nul 5255

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-in 3911  df-ss 3921  df-nul 4286  df-pw 4556  df-sn 4582  df-uni 4865

This theorem is referenced by:  mnuprdlem4  44815