Theorem mnuop123d

Description: Operations of a minimal universe. (Contributed by Rohan Ridenour, 13-Aug-2023)

	Ref	Expression
Hypotheses	mnuop123d.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 ) ) ) ) }
	mnuop123d.2	⊢ ( 𝜑 → 𝑈 ∈ 𝑀 )
	mnuop123d.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
Assertion	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 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		mnuop123d.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		mnuop123d.2	⊢ ( 𝜑 → 𝑈 ∈ 𝑀 )
3		mnuop123d.3	⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
4		pweq	⊢ ( 𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴 )
5	4	sseq1d	⊢ ( 𝑧 = 𝐴 → ( 𝒫 𝑧 ⊆ 𝑈 ↔ 𝒫 𝐴 ⊆ 𝑈 ) )
6	4	sseq1d	⊢ ( 𝑧 = 𝐴 → ( 𝒫 𝑧 ⊆ 𝑤 ↔ 𝒫 𝐴 ⊆ 𝑤 ) )
7		raleq	⊢ ( 𝑧 = 𝐴 → ( ∀ 𝑖 ∈ 𝑧 ( ∃ 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 ) ) ) )
8	6 7	anbi12d	⊢ ( 𝑧 = 𝐴 → ( ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 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 ) ) ) ) )
9	8	rexbidv	⊢ ( 𝑧 = 𝐴 → ( ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 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 ) ) ) ) )
10	9	albidv	⊢ ( 𝑧 = 𝐴 → ( ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ↔ ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) )
11	5 10	anbi12d	⊢ ( 𝑧 = 𝐴 → ( ( 𝒫 𝑧 ⊆ 𝑈 ∧ ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ↔ ( 𝒫 𝐴 ⊆ 𝑈 ∧ ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ) )
12	1	ismnu	⊢ ( 𝑈 ∈ 𝑀 → ( 𝑈 ∈ 𝑀 ↔ ∀ 𝑧 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑈 ∧ ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ) )
13	12	ibi	⊢ ( 𝑈 ∈ 𝑀 → ∀ 𝑧 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑈 ∧ ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) )
14	2 13	syl	⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑈 ∧ ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) )
15	11 14 3	rspcdva	⊢ ( 𝜑 → ( 𝒫 𝐴 ⊆ 𝑈 ∧ ∀ 𝑓 ∃ 𝑤 ∈ 𝑈 ( 𝒫 𝐴 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝐴 ( ∃ 𝑣 ∈ 𝑈 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) )