Theorem pitonn 6924

Description: Mapping from N to ℕ. (Contributed by Jim Kingdon, 22-Apr-2020.)

Assertion

Ref	Expression
pitonn	⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})

Distinct variable groups:   𝑁,𝑙,𝑢   𝑦,𝑙,𝑢   𝑥,𝑦

Allowed substitution hints:   𝑁(𝑥,𝑦)

Proof of Theorem pitonn

Dummy variables 𝑤 𝑧 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3549	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 1𝑜 → ⟨𝑤, 1𝑜⟩ = ⟨1𝑜, 1𝑜⟩)
2	1	eceq1d 6142	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 1𝑜 → [⟨𝑤, 1𝑜⟩] ~Q = [⟨1𝑜, 1𝑜⟩] ~Q )
3	2	breq2d 3776	. . . . . . . . . . . . 13 ⊢ (𝑤 = 1𝑜 → (𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q ↔ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q ))
4	3	abbidv 2155	. . . . . . . . . . . 12 ⊢ (𝑤 = 1𝑜 → {𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q })
5	2	breq1d 3774	. . . . . . . . . . . . 13 ⊢ (𝑤 = 1𝑜 → ([⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢))
6	5	abbidv 2155	. . . . . . . . . . . 12 ⊢ (𝑤 = 1𝑜 → {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢})
7	4, 6	opeq12d 3557	. . . . . . . . . . 11 ⊢ (𝑤 = 1𝑜 → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩)
8	7	oveq1d 5527	. . . . . . . . . 10 ⊢ (𝑤 = 1𝑜 → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P))
9	8	opeq1d 3555	. . . . . . . . 9 ⊢ (𝑤 = 1𝑜 → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
10	9	eceq1d 6142	. . . . . . . 8 ⊢ (𝑤 = 1𝑜 → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
11	10	opeq1d 3555	. . . . . . 7 ⊢ (𝑤 = 1𝑜 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
12	11	eleq1d 2106	. . . . . 6 ⊢ (𝑤 = 1𝑜 → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
13	12	imbi2d 219	. . . . 5 ⊢ (𝑤 = 1𝑜 → (((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧) ↔ ((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)))
14		opeq1 3549	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑘 → ⟨𝑤, 1𝑜⟩ = ⟨𝑘, 1𝑜⟩)
15	14	eceq1d 6142	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑘 → [⟨𝑤, 1𝑜⟩] ~Q = [⟨𝑘, 1𝑜⟩] ~Q )
16	15	breq2d 3776	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑘 → (𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q ↔ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q ))
17	16	abbidv 2155	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑘 → {𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q })
18	15	breq1d 3774	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑘 → ([⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢))
19	18	abbidv 2155	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑘 → {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢})
20	17, 19	opeq12d 3557	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑘 → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩)
21	20	oveq1d 5527	. . . . . . . . . 10 ⊢ (𝑤 = 𝑘 → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P))
22	21	opeq1d 3555	. . . . . . . . 9 ⊢ (𝑤 = 𝑘 → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
23	22	eceq1d 6142	. . . . . . . 8 ⊢ (𝑤 = 𝑘 → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
24	23	opeq1d 3555	. . . . . . 7 ⊢ (𝑤 = 𝑘 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
25	24	eleq1d 2106	. . . . . 6 ⊢ (𝑤 = 𝑘 → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
26	25	imbi2d 219	. . . . 5 ⊢ (𝑤 = 𝑘 → (((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧) ↔ ((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)))
27		opeq1 3549	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑘 +N 1𝑜) → ⟨𝑤, 1𝑜⟩ = ⟨(𝑘 +N 1𝑜), 1𝑜⟩)
28	27	eceq1d 6142	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑘 +N 1𝑜) → [⟨𝑤, 1𝑜⟩] ~Q = [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q )
29	28	breq2d 3776	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑘 +N 1𝑜) → (𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q ↔ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q ))
30	29	abbidv 2155	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑘 +N 1𝑜) → {𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q })
31	28	breq1d 3774	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑘 +N 1𝑜) → ([⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢))
32	31	abbidv 2155	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑘 +N 1𝑜) → {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢})
33	30, 32	opeq12d 3557	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑘 +N 1𝑜) → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩)
34	33	oveq1d 5527	. . . . . . . . . 10 ⊢ (𝑤 = (𝑘 +N 1𝑜) → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P))
35	34	opeq1d 3555	. . . . . . . . 9 ⊢ (𝑤 = (𝑘 +N 1𝑜) → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
36	35	eceq1d 6142	. . . . . . . 8 ⊢ (𝑤 = (𝑘 +N 1𝑜) → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
37	36	opeq1d 3555	. . . . . . 7 ⊢ (𝑤 = (𝑘 +N 1𝑜) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
38	37	eleq1d 2106	. . . . . 6 ⊢ (𝑤 = (𝑘 +N 1𝑜) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
39	38	imbi2d 219	. . . . 5 ⊢ (𝑤 = (𝑘 +N 1𝑜) → (((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧) ↔ ((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)))
40		opeq1 3549	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑁 → ⟨𝑤, 1𝑜⟩ = ⟨𝑁, 1𝑜⟩)
41	40	eceq1d 6142	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑁 → [⟨𝑤, 1𝑜⟩] ~Q = [⟨𝑁, 1𝑜⟩] ~Q )
42	41	breq2d 3776	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑁 → (𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q ↔ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q ))
43	42	abbidv 2155	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑁 → {𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q })
44	41	breq1d 3774	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑁 → ([⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢))
45	44	abbidv 2155	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑁 → {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢})
46	43, 45	opeq12d 3557	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑁 → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩)
47	46	oveq1d 5527	. . . . . . . . . 10 ⊢ (𝑤 = 𝑁 → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P))
48	47	opeq1d 3555	. . . . . . . . 9 ⊢ (𝑤 = 𝑁 → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
49	48	eceq1d 6142	. . . . . . . 8 ⊢ (𝑤 = 𝑁 → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
50	49	opeq1d 3555	. . . . . . 7 ⊢ (𝑤 = 𝑁 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
51	50	eleq1d 2106	. . . . . 6 ⊢ (𝑤 = 𝑁 → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
52	51	imbi2d 219	. . . . 5 ⊢ (𝑤 = 𝑁 → (((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧) ↔ ((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)))
53		pitonnlem1 6921	. . . . . . . 8 ⊢ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
54	53	eleq1i 2103	. . . . . . 7 ⊢ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ 1 ∈ 𝑧)
55	54	biimpri 124	. . . . . 6 ⊢ (1 ∈ 𝑧 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)
56	55	adantr 261	. . . . 5 ⊢ ((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1𝑜, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨1𝑜, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)
57		oveq1 5519	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → (𝑦 + 1) = (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1))
58	57	eleq1d 2106	. . . . . . . . . 10 ⊢ (𝑦 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ((𝑦 + 1) ∈ 𝑧 ↔ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) ∈ 𝑧))
59	58	rspccv 2653	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧 → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) ∈ 𝑧))
60	59	ad2antll 460	. . . . . . . 8 ⊢ ((𝑘 ∈ N ∧ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) ∈ 𝑧))
61		pitonnlem2 6923	. . . . . . . . . 10 ⊢ (𝑘 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
62	61	eleq1d 2106	. . . . . . . . 9 ⊢ (𝑘 ∈ N → ((⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) ∈ 𝑧 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
63	62	adantr 261	. . . . . . . 8 ⊢ ((𝑘 ∈ N ∧ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) → ((⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) ∈ 𝑧 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
64	60, 63	sylibd 138	. . . . . . 7 ⊢ ((𝑘 ∈ N ∧ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
65	64	ex 108	. . . . . 6 ⊢ (𝑘 ∈ N → ((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)))
66	65	a2d 23	. . . . 5 ⊢ (𝑘 ∈ N → (((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑘, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑘, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧) → ((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨(𝑘 +N 1𝑜), 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)))
67	13, 26, 39, 52, 56, 66	indpi 6440	. . . 4 ⊢ (𝑁 ∈ N → ((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
68	67	alrimiv 1754	. . 3 ⊢ (𝑁 ∈ N → ∀𝑧((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
69		eleq2 2101	. . . . 5 ⊢ (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧))
70		eleq2 2101	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧))
71	70	raleqbi1dv 2513	. . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧))
72	69, 71	anbi12d 442	. . . 4 ⊢ (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)))
73	72	ralab 2701	. . 3 ⊢ (∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ ∀𝑧((1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
74	68, 73	sylibr 137	. 2 ⊢ (𝑁 ∈ N → ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)
75		nnprlu 6651	. . . . . . 7 ⊢ (𝑁 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
76		1pr 6652	. . . . . . 7 ⊢ 1P ∈ P
77		addclpr 6635	. . . . . . 7 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P ∧ 1P ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
78	75, 76, 77	sylancl 392	. . . . . 6 ⊢ (𝑁 ∈ N → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
79		opelxpi 4376	. . . . . 6 ⊢ (((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1P ∈ P) → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
80	78, 76, 79	sylancl 392	. . . . 5 ⊢ (𝑁 ∈ N → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
81		enrex 6822	. . . . . 6 ⊢ ~R ∈ V
82	81	ecelqsi 6160	. . . . 5 ⊢ (⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P) → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ ((P × P) / ~R ))
83	80, 82	syl 14	. . . 4 ⊢ (𝑁 ∈ N → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ ((P × P) / ~R ))
84		0r 6835	. . . 4 ⊢ 0R ∈ R
85		opelxpi 4376	. . . 4 ⊢ (([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ ((P × P) / ~R ) ∧ 0R ∈ R) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ (((P × P) / ~R ) × R))
86	83, 84, 85	sylancl 392	. . 3 ⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ (((P × P) / ~R ) × R))
87		elintg 3623	. . 3 ⊢ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ (((P × P) / ~R ) × R) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
88	86, 87	syl 14	. 2 ⊢ (𝑁 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)} ↔ ∀𝑧 ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧))
89	74, 88	mpbird 156	1 ⊢ (𝑁 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ⟨cop 3378  ∩ cint 3615   class class class wbr 3764   × cxp 4343  (class class class)co 5512  1_𝑜c1o 5994  [cec 6104   / cqs 6105  Ncnpi 6370   +_N cpli 6371   ~_Q ceq 6377   <_Q cltq 6383  Pcnp 6389  1_Pc1p 6390   +_P cpp 6391   ~_R cer 6394  Rcnr 6395  0_Rc0r 6396  1c1 6890   + caddc 6892

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-iplp 6566  df-enr 6811  df-nr 6812  df-plr 6813  df-0r 6816  df-1r 6817  df-c 6895  df-1 6897  df-add 6900

This theorem is referenced by:  axarch  6965  axcaucvglemcl  6969  axcaucvglemval  6971  axcaucvglemcau  6972  axcaucvglemres  6973