Theorem pitonn 7810

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 3765	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 1o → ⟨𝑤, 1o⟩ = ⟨1o, 1o⟩)
2	1	eceq1d 6549	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 1o → [⟨𝑤, 1o⟩] ~Q = [⟨1o, 1o⟩] ~Q )
3	2	breq2d 4001	. . . . . . . . . . . . 13 ⊢ (𝑤 = 1o → (𝑙 <Q [⟨𝑤, 1o⟩] ~Q ↔ 𝑙 <Q [⟨1o, 1o⟩] ~Q ))
4	3	abbidv 2288	. . . . . . . . . . . 12 ⊢ (𝑤 = 1o → {𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q })
5	2	breq1d 3999	. . . . . . . . . . . . 13 ⊢ (𝑤 = 1o → ([⟨𝑤, 1o⟩] ~Q <Q 𝑢 ↔ [⟨1o, 1o⟩] ~Q <Q 𝑢))
6	5	abbidv 2288	. . . . . . . . . . . 12 ⊢ (𝑤 = 1o → {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢})
7	4, 6	opeq12d 3773	. . . . . . . . . . 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 5868	. . . . . . . . . 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 3771	. . . . . . . . 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 6549	. . . . . . . 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 3771	. . . . . . 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 2239	. . . . . 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 229	. . . . 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 3765	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑘 → ⟨𝑤, 1o⟩ = ⟨𝑘, 1o⟩)
15	14	eceq1d 6549	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑘 → [⟨𝑤, 1o⟩] ~Q = [⟨𝑘, 1o⟩] ~Q )
16	15	breq2d 4001	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑘 → (𝑙 <Q [⟨𝑤, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q ))
17	16	abbidv 2288	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑘 → {𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑘, 1o⟩] ~Q })
18	15	breq1d 3999	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑘 → ([⟨𝑤, 1o⟩] ~Q <Q 𝑢 ↔ [⟨𝑘, 1o⟩] ~Q <Q 𝑢))
19	18	abbidv 2288	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑘 → {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑘, 1o⟩] ~Q <Q 𝑢})
20	17, 19	opeq12d 3773	. . . . . . . . . . 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 5868	. . . . . . . . . 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 3771	. . . . . . . . 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 6549	. . . . . . . 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 3771	. . . . . . 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 2239	. . . . . 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 229	. . . . 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 3765	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑘 +N 1o) → ⟨𝑤, 1o⟩ = ⟨(𝑘 +N 1o), 1o⟩)
28	27	eceq1d 6549	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑘 +N 1o) → [⟨𝑤, 1o⟩] ~Q = [⟨(𝑘 +N 1o), 1o⟩] ~Q )
29	28	breq2d 4001	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑘 +N 1o) → (𝑙 <Q [⟨𝑤, 1o⟩] ~Q ↔ 𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q ))
30	29	abbidv 2288	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑘 +N 1o) → {𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨(𝑘 +N 1o), 1o⟩] ~Q })
31	28	breq1d 3999	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑘 +N 1o) → ([⟨𝑤, 1o⟩] ~Q <Q 𝑢 ↔ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢))
32	31	abbidv 2288	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑘 +N 1o) → {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨(𝑘 +N 1o), 1o⟩] ~Q <Q 𝑢})
33	30, 32	opeq12d 3773	. . . . . . . . . . 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 5868	. . . . . . . . . 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 3771	. . . . . . . . 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 6549	. . . . . . . 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 3771	. . . . . . 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 2239	. . . . . 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 229	. . . . 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 3765	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑁 → ⟨𝑤, 1o⟩ = ⟨𝑁, 1o⟩)
41	40	eceq1d 6549	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑁 → [⟨𝑤, 1o⟩] ~Q = [⟨𝑁, 1o⟩] ~Q )
42	41	breq2d 4001	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑁 → (𝑙 <Q [⟨𝑤, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q ))
43	42	abbidv 2288	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑁 → {𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q })
44	41	breq1d 3999	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑁 → ([⟨𝑤, 1o⟩] ~Q <Q 𝑢 ↔ [⟨𝑁, 1o⟩] ~Q <Q 𝑢))
45	44	abbidv 2288	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑁 → {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢})
46	43, 45	opeq12d 3773	. . . . . . . . . . 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 5868	. . . . . . . . . 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 3771	. . . . . . . . 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 6549	. . . . . . . 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 3771	. . . . . . 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 2239	. . . . . 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 229	. . . . 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 7807	. . . . . . . 8 ⊢ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = 1
54	53	eleq1i 2236	. . . . . . 7 ⊢ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧 ↔ 1 ∈ 𝑧)
55	54	biimpri 132	. . . . . 6 ⊢ (1 ∈ 𝑧 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨1o, 1o⟩] ~Q }, {𝑢 ∣ [⟨1o, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑧)
56	55	adantr 274	. . . . 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 5860	. . . . . . . . . . 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 2239	. . . . . . . . . 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 2831	. . . . . . . . 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 488	. . . . . . . 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 7809	. . . . . . . . . 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 2239	. . . . . . . . 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 274	. . . . . . . 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 148	. . . . . . 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 114	. . . . . 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 7304	. . . 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 1867	. . 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 2234	. . . . 5 ⊢ (𝑥 = 𝑧 → (1 ∈ 𝑥 ↔ 1 ∈ 𝑧))
70		eleq2 2234	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑦 + 1) ∈ 𝑥 ↔ (𝑦 + 1) ∈ 𝑧))
71	70	raleqbi1dv 2673	. . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧))
72	69, 71	anbi12d 470	. . . 4 ⊢ (𝑥 = 𝑧 → ((1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥) ↔ (1 ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (𝑦 + 1) ∈ 𝑧)))
73	72	ralab 2890	. . 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 133	. 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 7515	. . . . . . 7 ⊢ (𝑁 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
76		1pr 7516	. . . . . . 7 ⊢ 1P ∈ P
77		addclpr 7499	. . . . . . 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 411	. . . . . 6 ⊢ (𝑁 ∈ N → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
79		opelxpi 4643	. . . . . 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 411	. . . . 5 ⊢ (𝑁 ∈ N → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
81		enrex 7699	. . . . . 6 ⊢ ~R ∈ V
82	81	ecelqsi 6567	. . . . 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 7712	. . . 4 ⊢ 0R ∈ R
85		opelxpi 4643	. . . 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 411	. . 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 3839	. . 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 166	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 103   ↔ wb 104  ∀wal 1346   = wceq 1348   ∈ wcel 2141  {cab 2156  ∀wral 2448  ⟨cop 3586  ∩ cint 3831   class class class wbr 3989   × cxp 4609  (class class class)co 5853  1_oc1o 6388  [cec 6511   / cqs 6512  Ncnpi 7234   +_N cpli 7235   ~_Q ceq 7241   <_Q cltq 7247  Pcnp 7253  1_Pc1p 7254   +_P cpp 7255   ~_R cer 7258  Rcnr 7259  0_Rc0r 7260  1c1 7775   + caddc 7777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-i1p 7429  df-iplp 7430  df-enr 7688  df-nr 7689  df-plr 7690  df-0r 7693  df-1r 7694  df-c 7780  df-1 7782  df-add 7785

This theorem is referenced by:  axarch  7853  axcaucvglemcl  7857  axcaucvglemval  7859  axcaucvglemcau  7860  axcaucvglemres  7861