Theorem mnutrcld

Description: Minimal universes contain the elements of their elements. (Contributed by Rohan Ridenour, 13-Aug-2023)

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

Proof

Step	Hyp	Ref	Expression
1		mnutrcld.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		mnutrcld.2	⊢ ( 𝜑 → 𝑈 ∈ 𝑀 )
3		mnutrcld.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
4		mnutrcld.4	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
5	1 2 3	mnuunid	⊢ ( 𝜑 → ∪ 𝐴 ∈ 𝑈 )
6		elssuni	⊢ ( 𝐵 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝐴 )
7	4 6	syl	⊢ ( 𝜑 → 𝐵 ⊆ ∪ 𝐴 )
8	1 2 5 7	mnussd	⊢ ( 𝜑 → 𝐵 ∈ 𝑈 )