Theorem rr-grothprimbi

Description: Express "every set is contained in a Grothendieck universe" using only primitives. The right side (without the outermost universal quantifier) is proven as rr-grothprim . (Contributed by Rohan Ridenour, 13-Aug-2023)

	Ref	Expression
Assertion	rr-grothprimbi	⊢ ( ∀ 𝑥 ∃ 𝑦 ∈ Univ 𝑥 ∈ 𝑦 ↔ ∀ 𝑥 ¬ ∀ 𝑦 ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑓 ¬ ∀ 𝑤 ( 𝑤 ∈ 𝑦 → ¬ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑦 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-rex	⊢ ( ∃ 𝑦 ∈ Univ 𝑥 ∈ 𝑦 ↔ ∃ 𝑦 ( 𝑦 ∈ Univ ∧ 𝑥 ∈ 𝑦 ) )
2		ancom	⊢ ( ( 𝑦 ∈ Univ ∧ 𝑥 ∈ 𝑦 ) ↔ ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ Univ ) )
3		biid	⊢ ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦 )
4		grumnueq	⊢ Univ = { 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 ) ) ) ) }
5	4	ismnu	⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ Univ ↔ ∀ 𝑧 ∈ 𝑦 ( 𝒫 𝑧 ⊆ 𝑦 ∧ ∀ 𝑓 ∃ 𝑤 ∈ 𝑦 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑦 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ) )
6	5	elv	⊢ ( 𝑦 ∈ Univ ↔ ∀ 𝑧 ∈ 𝑦 ( 𝒫 𝑧 ⊆ 𝑦 ∧ ∀ 𝑓 ∃ 𝑤 ∈ 𝑦 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑦 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) )
7		ismnuprim	⊢ ( ∀ 𝑧 ∈ 𝑦 ( 𝒫 𝑧 ⊆ 𝑦 ∧ ∀ 𝑓 ∃ 𝑤 ∈ 𝑦 ( 𝒫 𝑧 ⊆ 𝑤 ∧ ∀ 𝑖 ∈ 𝑧 ( ∃ 𝑣 ∈ 𝑦 ( 𝑖 ∈ 𝑣 ∧ 𝑣 ∈ 𝑓 ) → ∃ 𝑢 ∈ 𝑓 ( 𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤 ) ) ) ) ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑓 ¬ ∀ 𝑤 ( 𝑤 ∈ 𝑦 → ¬ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑦 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) ) )
8	6 7	bitri	⊢ ( 𝑦 ∈ Univ ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑓 ¬ ∀ 𝑤 ( 𝑤 ∈ 𝑦 → ¬ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑦 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) ) )
9	3 8	expandan	⊢ ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ Univ ) ↔ ¬ ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑓 ¬ ∀ 𝑤 ( 𝑤 ∈ 𝑦 → ¬ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑦 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) ) ) )
10	2 9	bitri	⊢ ( ( 𝑦 ∈ Univ ∧ 𝑥 ∈ 𝑦 ) ↔ ¬ ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑓 ¬ ∀ 𝑤 ( 𝑤 ∈ 𝑦 → ¬ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑦 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) ) ) )
11	10	expandexn	⊢ ( ∃ 𝑦 ( 𝑦 ∈ Univ ∧ 𝑥 ∈ 𝑦 ) ↔ ¬ ∀ 𝑦 ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑓 ¬ ∀ 𝑤 ( 𝑤 ∈ 𝑦 → ¬ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑦 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) ) ) )
12	1 11	bitri	⊢ ( ∃ 𝑦 ∈ Univ 𝑥 ∈ 𝑦 ↔ ¬ ∀ 𝑦 ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑓 ¬ ∀ 𝑤 ( 𝑤 ∈ 𝑦 → ¬ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑦 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) ) ) )
13	12	albii	⊢ ( ∀ 𝑥 ∃ 𝑦 ∈ Univ 𝑥 ∈ 𝑦 ↔ ∀ 𝑥 ¬ ∀ 𝑦 ( 𝑥 ∈ 𝑦 → ¬ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → ∀ 𝑓 ¬ ∀ 𝑤 ( 𝑤 ∈ 𝑦 → ¬ ∀ 𝑣 ¬ ( ( ∀ 𝑡 ( 𝑡 ∈ 𝑣 → 𝑡 ∈ 𝑧 ) → ¬ ( 𝑣 ∈ 𝑦 → ¬ 𝑣 ∈ 𝑤 ) ) → ¬ ∀ 𝑖 ( 𝑖 ∈ 𝑧 → ( 𝑣 ∈ 𝑦 → ( 𝑖 ∈ 𝑣 → ( 𝑣 ∈ 𝑓 → ¬ ∀ 𝑢 ( 𝑢 ∈ 𝑓 → ( 𝑖 ∈ 𝑢 → ¬ ∀ 𝑜 ( 𝑜 ∈ 𝑢 → ∀ 𝑠 ( 𝑠 ∈ 𝑜 → 𝑠 ∈ 𝑤 ) ) ) ) ) ) ) ) ) ) ) ) )