Theorem pitonn 7968

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 3821	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 1o → ⟨𝑤, 1o⟩ = ⟨1o, 1o⟩)
2	1	eceq1d 6663	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 1o → [⟨𝑤, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
3	2	breq2d 4059	. . . . . . . . . . . . 13 ⊢ (𝑤 = 1o → (𝑙 <Q [⟨𝑤, 1o⟩] ~Q ↔ 𝑙 <Q [⟨1o, 1o⟩] ~Q ))
4	3	abbidv 2324	. . . . . . . . . . . 12 ⊢ (𝑤 = 1o → {𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q })
5	2	breq1d 4057	. . . . . . . . . . . . 13 ⊢ (𝑤 = 1o → ([⟨𝑤, 1o⟩] ~Q <Q 𝑢 ↔ [⟨1o, 1o⟩] ~Q <Q 𝑢))
6	5	abbidv 2324	. . . . . . . . . . . 12 ⊢ (𝑤 = 1o → {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢})
7	4, 6	opeq12d 3829	. . . . . . . . . . 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 5966	. . . . . . . . . 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 3827	. . . . . . . . 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 6663	. . . . . . . 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 3827	. . . . . . 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 2275	. . . . . 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 230	. . . . 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 3821	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑘 → ⟨𝑤, 1o⟩ = ⟨𝑘, 1o⟩)
15	14	eceq1d 6663	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑘 → [⟨𝑤, 1o⟩] ~Q = [⟨𝑘, 1o⟩] ~Q )
16	15	breq2d 4059	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑘 → (𝑙 <Q [⟨𝑤, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q ))
17	16	abbidv 2324	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑘 → {𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q })
18	15	breq1d 4057	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑘 → ([⟨𝑤, 1o⟩] ~Q <Q 𝑢 ↔ [⟨𝑘, 1o⟩] ~Q <Q 𝑢))
19	18	abbidv 2324	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑘 → {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢})
20	17, 19	opeq12d 3829	. . . . . . . . . . 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 5966	. . . . . . . . . 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 3827	. . . . . . . . 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 6663	. . . . . . . 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 3827	. . . . . . 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 2275	. . . . . 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 230	. . . . 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 3821	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑘 +N 1o) → ⟨𝑤, 1o⟩ = ⟨(𝑘 +N 1o), 1o⟩)
28	27	eceq1d 6663	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑘 +N 1o) → [⟨𝑤, 1o⟩] ~Q = [⟨(𝑘 +N 1o), 1o⟩] ~Q )
29	28	breq2d 4059	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑘 +N 1o) → (𝑙 <Q [⟨𝑤, 1o⟩] ~Q ↔ 𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q ))
30	29	abbidv 2324	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑘 +N 1o) → {𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q })
31	28	breq1d 4057	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑘 +N 1o) → ([⟨𝑤, 1o⟩] ~Q <Q 𝑢 ↔ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢))
32	31	abbidv 2324	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑘 +N 1o) → {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢})
33	30, 32	opeq12d 3829	. . . . . . . . . . 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 5966	. . . . . . . . . 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 3827	. . . . . . . . 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 6663	. . . . . . . 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 3827	. . . . . . 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 2275	. . . . . 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 230	. . . . 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 3821	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑁 → ⟨𝑤, 1o⟩ = ⟨𝑁, 1o⟩)
41	40	eceq1d 6663	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑁 → [⟨𝑤, 1o⟩] ~Q = [⟨𝑁, 1o⟩] ~Q )
42	41	breq2d 4059	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑁 → (𝑙 <Q [⟨𝑤, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q ))
43	42	abbidv 2324	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑁 → {𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q })
44	41	breq1d 4057	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑁 → ([⟨𝑤, 1o⟩] ~Q <Q 𝑢 ↔ [⟨𝑁, 1o⟩] ~Q <Q 𝑢))
45	44	abbidv 2324	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑁 → {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢})
46	43, 45	opeq12d 3829	. . . . . . . . . . 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 5966	. . . . . . . . . 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 3827	. . . . . . . . 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 6663	. . . . . . . 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 3827	. . . . . . 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 2275	. . . . . 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 230	. . . . 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 7965	. . . . . . . 8 ⊢ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
54	53	eleq1i 2272	. . . . . . 7 ⊢ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ 1 ∈ 𝑧)
55	54	biimpri 133	. . . . . 6 ⊢ (1 ∈ 𝑧 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)
56	55	adantr 276	. . . . 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 5958	. . . . . . . . . . 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 2275	. . . . . . . . . 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 2875	. . . . . . . . 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 491	. . . . . . . 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 7967	. . . . . . . . . 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 2275	. . . . . . . . 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 276	. . . . . . . 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 149	. . . . . . 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 115	. . . . . 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 26	. . . . 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 7462	. . . 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 1898	. . 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 2270	. . . . 5 ⊢ (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧))
70		eleq2 2270	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧))
71	70	raleqbi1dv 2715	. . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧))
72	69, 71	anbi12d 473	. . . 4 ⊢ (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)))
73	72	ralab 2934	. . 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 134	. 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 7673	. . . . . . 7 ⊢ (𝑁 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
76		1pr 7674	. . . . . . 7 ⊢ 1P ∈ P
77		addclpr 7657	. . . . . . 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 413	. . . . . 6 ⊢ (𝑁 ∈ N → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
79		opelxpi 4711	. . . . . 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 413	. . . . 5 ⊢ (𝑁 ∈ N → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
81		enrex 7857	. . . . . 6 ⊢ ~R ∈ V
82	81	ecelqsi 6683	. . . . 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 7870	. . . 4 ⊢ 0R ∈ R
85		opelxpi 4711	. . . 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 413	. . 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 3895	. . 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 167	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 104   ↔ wb 105  ∀wal 1371   = wceq 1373   ∈ wcel 2177  {cab 2192  ∀wral 2485  ⟨cop 3637  ∩ cint 3887   class class class wbr 4047   × cxp 4677  (class class class)co 5951  1_oc1o 6502  [cec 6625   / cqs 6626  Ncnpi 7392   +_N cpli 7393   ~_Q ceq 7399   <_Q cltq 7405  Pcnp 7411  1_Pc1p 7412   +_P cpp 7413   ~_R cer 7416  Rcnr 7417  0_Rc0r 7418  1c1 7933   + caddc 7935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-eprel 4340  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-1o 6509  df-2o 6510  df-oadd 6513  df-omul 6514  df-er 6627  df-ec 6629  df-qs 6633  df-ni 7424  df-pli 7425  df-mi 7426  df-lti 7427  df-plpq 7464  df-mpq 7465  df-enq 7467  df-nqqs 7468  df-plqqs 7469  df-mqqs 7470  df-1nqqs 7471  df-rq 7472  df-ltnqqs 7473  df-enq0 7544  df-nq0 7545  df-0nq0 7546  df-plq0 7547  df-mq0 7548  df-inp 7586  df-i1p 7587  df-iplp 7588  df-enr 7846  df-nr 7847  df-plr 7848  df-0r 7851  df-1r 7852  df-c 7938  df-1 7940  df-add 7943

This theorem is referenced by:  axarch  8011  axcaucvglemcl  8015  axcaucvglemval  8017  axcaucvglemcau  8018  axcaucvglemres  8019