Theorem mnuop23d

Description: Second and third operations of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023)

	Ref	Expression
Hypotheses	mnuop23d.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 ) ) ) ) }
	mnuop23d.2	⊢ ( 𝜑 → 𝑈 ∈ 𝑀 )
	mnuop23d.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
	mnuop23d.4	⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
Assertion	mnuop23d	⊢ ( 𝜑 → ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹 ) → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		mnuop23d.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		mnuop23d.2	⊢ ( 𝜑 → 𝑈 ∈ 𝑀 )
3		mnuop23d.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
4		mnuop23d.4	⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
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	simprd	⊢ ( 𝜑 → ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) )
7		eleq2	⊢ ( 𝑓 = 𝐹 → ( 𝑣 ∈ 𝑓 ↔ 𝑣 ∈ 𝐹 ) )
8	7	anbi2d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) ↔ ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹 ) ) )
9	8	rexbidv	⊢ ( 𝑓 = 𝐹 → ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) ↔ ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹 ) ) )
10		rexeq	⊢ ( 𝑓 = 𝐹 → ( ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ↔ ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) )
11	9 10	imbi12d	⊢ ( 𝑓 = 𝐹 → ( ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ↔ ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹 ) → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) )
12	11	ralbidv	⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ↔ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹 ) → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) )
13	12	anbi2d	⊢ ( 𝑓 = 𝐹 → ( ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ↔ ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹 ) → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) )
14	13	rexbidv	⊢ ( 𝑓 = 𝐹 → ( ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ↔ ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹 ) → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) )
15	14	spcgv	⊢ ( 𝐹 ∈ 𝑉 → ( ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) → ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹 ) → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) )
16	4 6 15	sylc	⊢ ( 𝜑 → ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝐹 ) → ∃ 𝑢 ∈ 𝐹 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) )