Theorem mnussd

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

	Ref	Expression
Hypotheses	mnussd.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 ) ) ) ) }
	mnussd.2	⊢ ( 𝜑 → 𝑈 ∈ 𝑀 )
	mnussd.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
	mnussd.4	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
Assertion	mnussd	⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		mnussd.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		mnussd.2	⊢ ( 𝜑 → 𝑈 ∈ 𝑀 )
3		mnussd.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
4		mnussd.4	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
5	1 2 3	mnuop123d	⊢ ( 𝜑 → ( 𝒫 𝐴 ⊆ 𝑈 ∧ ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) )
6	5	simpld	⊢ ( 𝜑 → 𝒫 𝐴 ⊆ 𝑈 )
7	3 4	sselpwd	⊢ ( 𝜑 → 𝐵 ∈ 𝒫 𝐴 )
8	6 7	sseldd	⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )