Theorem ballotlem7 34517

Description: 𝑅 is a bijection between two subsets of (𝑂 ∖ 𝐸): one where a vote for A is picked first, and one where a vote for B is picked first. (Contributed by Thierry Arnoux, 12-Dec-2016.)

Hypotheses

Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotth.o	⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
ballotth.p	⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
ballotth.f	⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
ballotth.e	⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
ballotth.mgtn	⊢ 𝑁 < 𝑀
ballotth.i	⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
ballotth.s	⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
ballotth.r	⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))

Assertion

Ref	Expression
ballotlem7	⊢ (𝑅 ↾ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}):{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}–1-1-onto→{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}

Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐   𝑖,𝑀   𝑖,𝑁   𝑖,𝑂   𝑘,𝑀   𝑘,𝑁   𝑘,𝑂   𝑖,𝑐,𝐹,𝑘   𝑖,𝐸,𝑘   𝑘,𝐼,𝑐   𝐸,𝑐   𝑖,𝐼,𝑐   𝑆,𝑘,𝑖,𝑐   𝑅,𝑖,𝑘   𝑥,𝑐,𝐹   𝑥,𝑀   𝑥,𝑁,𝑘,𝑖

Allowed substitution hints:   𝑃(𝑥,𝑖,𝑘,𝑐)   𝑅(𝑥,𝑐)   𝑆(𝑥)   𝐸(𝑥)   𝐼(𝑥)   𝑂(𝑥)

Proof of Theorem ballotlem7

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.r	. . 3 ⊢ 𝑅 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
2	1	funmpt2 6607	. 2 ⊢ Fun 𝑅
3		ballotth.m	. . 3 ⊢ 𝑀 ∈ ℕ
4		ballotth.n	. . 3 ⊢ 𝑁 ∈ ℕ
5		ballotth.o	. . 3 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
6		ballotth.p	. . 3 ⊢ 𝑃 = (𝑥 ∈ 𝒫 𝑂 ↦ ((♯‘𝑥) / (♯‘𝑂)))
7		ballotth.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝑂 ↦ (𝑖 ∈ ℤ ↦ ((♯‘((1...𝑖) ∩ 𝑐)) − (♯‘((1...𝑖) ∖ 𝑐)))))
8		ballotth.e	. . 3 ⊢ 𝐸 = {𝑐 ∈ 𝑂 ∣ ∀𝑖 ∈ (1...(𝑀 + 𝑁))0 < ((𝐹‘𝑐)‘𝑖)}
9		ballotth.mgtn	. . 3 ⊢ 𝑁 < 𝑀
10		ballotth.i	. . 3 ⊢ 𝐼 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ inf({𝑘 ∈ (1...(𝑀 + 𝑁)) ∣ ((𝐹‘𝑐)‘𝑘) = 0}, ℝ, < ))
11		ballotth.s	. . 3 ⊢ 𝑆 = (𝑐 ∈ (𝑂 ∖ 𝐸) ↦ (𝑖 ∈ (1...(𝑀 + 𝑁)) ↦ if(𝑖 ≤ (𝐼‘𝑐), (((𝐼‘𝑐) + 1) − 𝑖), 𝑖)))
12	3, 4, 5, 6, 7, 8, 9, 10, 11, 1	ballotlemrinv 34515	. 2 ⊢ ◡𝑅 = 𝑅
13		rabid 3455	. . . . . 6 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐))
14	3, 4, 5, 6, 7, 8, 9, 10, 11, 1	ballotlemrc 34512	. . . . . . . 8 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸))
15	14	adantr 480	. . . . . . 7 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐) → (𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸))
16	3, 4, 5, 6, 7, 8, 9, 10	ballotlem1c 34489	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐) → ¬ (𝐼‘𝑐) ∈ 𝑐)
17	16	ex 412	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (1 ∈ 𝑐 → ¬ (𝐼‘𝑐) ∈ 𝑐))
18	3, 4, 5, 6, 7, 8, 9, 10, 11, 1	ballotlem1ri 34516	. . . . . . . . . 10 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (1 ∈ (𝑅‘𝑐) ↔ (𝐼‘𝑐) ∈ 𝑐))
19	18	notbid 318	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (¬ 1 ∈ (𝑅‘𝑐) ↔ ¬ (𝐼‘𝑐) ∈ 𝑐))
20	17, 19	sylibrd 259	. . . . . . . 8 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (1 ∈ 𝑐 → ¬ 1 ∈ (𝑅‘𝑐)))
21	20	imp 406	. . . . . . 7 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐) → ¬ 1 ∈ (𝑅‘𝑐))
22	15, 21	jca 511	. . . . . 6 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐) → ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)))
23	13, 22	sylbi 217	. . . . 5 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} → ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)))
24	23	rgen 3061	. . . 4 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐))
25		eleq2 2828	. . . . . . . 8 ⊢ (𝑏 = (𝑅‘𝑐) → (1 ∈ 𝑏 ↔ 1 ∈ (𝑅‘𝑐)))
26	25	notbid 318	. . . . . . 7 ⊢ (𝑏 = (𝑅‘𝑐) → (¬ 1 ∈ 𝑏 ↔ ¬ 1 ∈ (𝑅‘𝑐)))
27	26	elrab 3695	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑏} ↔ ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)))
28		eleq2 2828	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (1 ∈ 𝑏 ↔ 1 ∈ 𝑐))
29	28	notbid 318	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (¬ 1 ∈ 𝑏 ↔ ¬ 1 ∈ 𝑐))
30	29	cbvrabv 3444	. . . . . . 7 ⊢ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
31	30	eleq2i 2831	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑏} ↔ (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
32	27, 31	bitr3i 277	. . . . 5 ⊢ (((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)) ↔ (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
33	32	ralbii 3091	. . . 4 ⊢ (∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)) ↔ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
34	24, 33	mpbi 230	. . 3 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
35		ssrab2 4090	. . . . 5 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
36		fvex 6920	. . . . . . 7 ⊢ (𝑆‘𝑐) ∈ V
37		imaexg 7936	. . . . . . 7 ⊢ ((𝑆‘𝑐) ∈ V → ((𝑆‘𝑐) “ 𝑐) ∈ V)
38	36, 37	ax-mp 5	. . . . . 6 ⊢ ((𝑆‘𝑐) “ 𝑐) ∈ V
39	38, 1	dmmpti 6713	. . . . 5 ⊢ dom 𝑅 = (𝑂 ∖ 𝐸)
40	35, 39	sseqtrri 4033	. . . 4 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ dom 𝑅
41		nfrab1 3454	. . . . 5 ⊢ Ⅎ𝑐{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
42		nfrab1 3454	. . . . 5 ⊢ Ⅎ𝑐{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
43		nfmpt1 5256	. . . . . 6 ⊢ Ⅎ𝑐(𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
44	1, 43	nfcxfr 2901	. . . . 5 ⊢ Ⅎ𝑐𝑅
45	41, 42, 44	funimass4f 32654	. . . 4 ⊢ ((Fun 𝑅 ∧ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ dom 𝑅) → ((𝑅 “ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}))
46	2, 40, 45	mp2an 692	. . 3 ⊢ ((𝑅 “ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
47	34, 46	mpbir 231	. 2 ⊢ (𝑅 “ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
48		rabid 3455	. . . . . 6 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐))
49	14	adantr 480	. . . . . . 7 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) → (𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸))
50	3, 4, 5, 6, 7, 8, 9, 10	ballotlemic 34488	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) → (𝐼‘𝑐) ∈ 𝑐)
51	50	ex 412	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (¬ 1 ∈ 𝑐 → (𝐼‘𝑐) ∈ 𝑐))
52	51, 18	sylibrd 259	. . . . . . . 8 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (¬ 1 ∈ 𝑐 → 1 ∈ (𝑅‘𝑐)))
53	52	imp 406	. . . . . . 7 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) → 1 ∈ (𝑅‘𝑐))
54	49, 53	jca 511	. . . . . 6 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) → ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)))
55	48, 54	sylbi 217	. . . . 5 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} → ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)))
56	55	rgen 3061	. . . 4 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐))
57	25	elrab 3695	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑏} ↔ ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)))
58	28	cbvrabv 3444	. . . . . . 7 ⊢ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
59	58	eleq2i 2831	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑏} ↔ (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐})
60	57, 59	bitr3i 277	. . . . 5 ⊢ (((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)) ↔ (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐})
61	60	ralbii 3091	. . . 4 ⊢ (∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)) ↔ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐})
62	56, 61	mpbi 230	. . 3 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
63		ssrab2 4090	. . . . 5 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
64	63, 39	sseqtrri 4033	. . . 4 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ dom 𝑅
65	42, 41, 44	funimass4f 32654	. . . 4 ⊢ ((Fun 𝑅 ∧ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ dom 𝑅) → ((𝑅 “ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ↔ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}))
66	2, 64, 65	mp2an 692	. . 3 ⊢ ((𝑅 “ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ↔ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐})
67	62, 66	mpbir 231	. 2 ⊢ (𝑅 “ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
68	2, 12, 47, 67, 40, 64	rinvf1o 32647	1 ⊢ (𝑅 ↾ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}):{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}–1-1-onto→{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433  Vcvv 3478   ∖ cdif 3960   ∩ cin 3962   ⊆ wss 3963  ifcif 4531  𝒫 cpw 4605   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5689   ↾ cres 5691   “ cima 5692  Fun wfun 6557  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431  infcinf 9479  ℝcr 11152  0cc0 11153  1c1 11154   + caddc 11156   < clt 11293   ≤ cle 11294   − cmin 11490   / cdiv 11918  ℕcn 12264  ℤcz 12611  ...cfz 13544  ♯chash 14366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-hash 14367

This theorem is referenced by:  ballotlem8  34518