Theorem pitonnlem2 7907

Description: Lemma for pitonn 7908. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.)

Assertion

Ref	Expression
pitonnlem2	⊢ (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⟩)

Distinct variable group:   𝐾,𝑙,𝑢

Proof of Theorem pitonnlem2

Step	Hyp	Ref	Expression
1		df-1 7880	. . . 4 ⊢ 1 = ⟨1R, 0R⟩
2	1	oveq2i 5929	. . 3 ⊢ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + ⟨1R, 0R⟩)
3		nnprlu 7613	. . . . . . . 8 ⊢ (𝐾 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
4		1pr 7614	. . . . . . . 8 ⊢ 1P ∈ P
5		addclpr 7597	. . . . . . . 8 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P ∧ 1P ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
6	3, 4, 5	sylancl 413	. . . . . . 7 ⊢ (𝐾 ∈ N → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
7		opelxpi 4691	. . . . . . 7 ⊢ (((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1P ∈ P) → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
8	6, 4, 7	sylancl 413	. . . . . 6 ⊢ (𝐾 ∈ N → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P))
9		enrex 7797	. . . . . . 7 ⊢ ~R ∈ V
10	9	ecelqsi 6643	. . . . . 6 ⊢ (⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ ∈ (P × P) → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ ((P × P) / ~R ))
11	8, 10	syl 14	. . . . 5 ⊢ (𝐾 ∈ N → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ ((P × P) / ~R ))
12		df-nr 7787	. . . . 5 ⊢ R = ((P × P) / ~R )
13	11, 12	eleqtrrdi 2287	. . . 4 ⊢ (𝐾 ∈ N → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R)
14		1sr 7811	. . . 4 ⊢ 1R ∈ R
15		addresr 7897	. . . 4 ⊢ (([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ∈ R ∧ 1R ∈ R) → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + ⟨1R, 0R⟩) = ⟨([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R), 0R⟩)
16	13, 14, 15	sylancl 413	. . 3 ⊢ (𝐾 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + ⟨1R, 0R⟩) = ⟨([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R), 0R⟩)
17	2, 16	eqtrid 2238	. 2 ⊢ (𝐾 ∈ N → (⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ + 1) = ⟨([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R), 0R⟩)
18		pitonnlem1p1 7906	. . . . 5 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P → [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P), 1P⟩] ~R )
19	6, 18	syl 14	. . . 4 ⊢ (𝐾 ∈ N → [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P), 1P⟩] ~R )
20		df-1r 7792	. . . . . 6 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
21	20	oveq2i 5929	. . . . 5 ⊢ ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R) = ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R )
22		addclpr 7597	. . . . . . . 8 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
23	4, 4, 22	mp2an 426	. . . . . . 7 ⊢ (1P +P 1P) ∈ P
24		addsrpr 7805	. . . . . . . 8 ⊢ ((((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1P ∈ P) ∧ ((1P +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
25	4, 24	mpanl2 435	. . . . . . 7 ⊢ (((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ ((1P +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
26	23, 4, 25	mpanr12 439	. . . . . 6 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P → ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
27	6, 26	syl 14	. . . . 5 ⊢ (𝐾 ∈ N → ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R [⟨(1P +P 1P), 1P⟩] ~R ) = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
28	21, 27	eqtrid 2238	. . . 4 ⊢ (𝐾 ∈ N → ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R) = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P (1P +P 1P)), (1P +P 1P)⟩] ~R )
29		addpinq1 7524	. . . . . . . . . . 11 ⊢ (𝐾 ∈ N → [⟨(𝐾 +N 1o), 1o⟩] ~Q = ([⟨𝐾, 1o⟩] ~Q +Q 1Q))
30	29	breq2d 4041	. . . . . . . . . 10 ⊢ (𝐾 ∈ N → (𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q ↔ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)))
31	30	abbidv 2311	. . . . . . . . 9 ⊢ (𝐾 ∈ N → {𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)})
32	29	breq1d 4039	. . . . . . . . . 10 ⊢ (𝐾 ∈ N → ([⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢 ↔ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢))
33	32	abbidv 2311	. . . . . . . . 9 ⊢ (𝐾 ∈ N → {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢})
34	31, 33	opeq12d 3812	. . . . . . . 8 ⊢ (𝐾 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)}, {𝑢 ∣ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢}⟩)
35		nnnq 7482	. . . . . . . . 9 ⊢ (𝐾 ∈ N → [⟨𝐾, 1o⟩] ~Q ∈ Q)
36		addnqpr1 7622	. . . . . . . . 9 ⊢ ([⟨𝐾, 1o⟩] ~Q ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)}, {𝑢 ∣ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
37	35, 36	syl 14	. . . . . . . 8 ⊢ (𝐾 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q ([⟨𝐾, 1o⟩] ~Q +Q 1Q)}, {𝑢 ∣ ([⟨𝐾, 1o⟩] ~Q +Q 1Q) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
38	34, 37	eqtrd 2226	. . . . . . 7 ⊢ (𝐾 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
39	38	oveq1d 5933	. . . . . 6 ⊢ (𝐾 ∈ N → (⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P))
40	39	opeq1d 3810	. . . . 5 ⊢ (𝐾 ∈ N → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P), 1P⟩)
41	40	eceq1d 6623	. . . 4 ⊢ (𝐾 ∈ N → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨((⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) +P 1P), 1P⟩] ~R )
42	19, 28, 41	3eqtr4d 2236	. . 3 ⊢ (𝐾 ∈ N → ([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R) = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
43	42	opeq1d 3810	. 2 ⊢ (𝐾 ∈ N → ⟨([⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐾, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐾, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R +R 1R), 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨(𝐾 +N 1o), 1o⟩] ~Q }, {𝑢 ∣ [⟨(𝐾 +N 1o), 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
44	17, 43	eqtrd 2226	1 ⊢ (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⟩)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  {cab 2179  ⟨cop 3621   class class class wbr 4029   × cxp 4657  (class class class)co 5918  1_oc1o 6462  [cec 6585   / cqs 6586  Ncnpi 7332   +_N cpli 7333   ~_Q ceq 7339  Qcnq 7340  1_Qc1q 7341   +_Q cplq 7342   <_Q cltq 7345  Pcnp 7351  1_Pc1p 7352   +_P cpp 7353   ~_R cer 7356  Rcnr 7357  0_Rc0r 7358  1_Rc1r 7359   +_R cplr 7361  1c1 7873   + caddc 7875

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-i1p 7527  df-iplp 7528  df-enr 7786  df-nr 7787  df-plr 7788  df-0r 7791  df-1r 7792  df-c 7878  df-1 7880  df-add 7883

This theorem is referenced by:  pitonn  7908  nntopi  7954