Theorem mnusnd

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

	Ref	Expression
Hypotheses	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	mnusnd	⊢ ( 𝜑 → { 𝐴 } ∈ 𝑈 )

Proof

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