Theorem ballotlem7 30597

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 5927	. 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 30595	. 2 ⊢ ◡𝑅 = 𝑅
13		rabid 3116	. . . . . 6 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐))
14	3, 4, 5, 6, 7, 8, 9, 10, 11, 1	ballotlemrc 30592	. . . . . . . 8 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸))
15	14	adantr 481	. . . . . . 7 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐) → (𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸))
16	3, 4, 5, 6, 7, 8, 9, 10	ballotlem1c 30569	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐) → ¬ (𝐼‘𝑐) ∈ 𝑐)
17	16	ex 450	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (1 ∈ 𝑐 → ¬ (𝐼‘𝑐) ∈ 𝑐))
18	3, 4, 5, 6, 7, 8, 9, 10, 11, 1	ballotlem1ri 30596	. . . . . . . . . 10 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (1 ∈ (𝑅‘𝑐) ↔ (𝐼‘𝑐) ∈ 𝑐))
19	18	notbid 308	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (¬ 1 ∈ (𝑅‘𝑐) ↔ ¬ (𝐼‘𝑐) ∈ 𝑐))
20	17, 19	sylibrd 249	. . . . . . . 8 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (1 ∈ 𝑐 → ¬ 1 ∈ (𝑅‘𝑐)))
21	20	imp 445	. . . . . . 7 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐) → ¬ 1 ∈ (𝑅‘𝑐))
22	15, 21	jca 554	. . . . . 6 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐) → ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)))
23	13, 22	sylbi 207	. . . . 5 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} → ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)))
24	23	rgen 2922	. . . 4 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐))
25		eleq2 2690	. . . . . . . 8 ⊢ (𝑏 = (𝑅‘𝑐) → (1 ∈ 𝑏 ↔ 1 ∈ (𝑅‘𝑐)))
26	25	notbid 308	. . . . . . 7 ⊢ (𝑏 = (𝑅‘𝑐) → (¬ 1 ∈ 𝑏 ↔ ¬ 1 ∈ (𝑅‘𝑐)))
27	26	elrab 3363	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑏} ↔ ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)))
28		eleq2 2690	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (1 ∈ 𝑏 ↔ 1 ∈ 𝑐))
29	28	notbid 308	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (¬ 1 ∈ 𝑏 ↔ ¬ 1 ∈ 𝑐))
30	29	cbvrabv 3199	. . . . . . 7 ⊢ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
31	30	eleq2i 2693	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑏} ↔ (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
32	27, 31	bitr3i 266	. . . . 5 ⊢ (((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)) ↔ (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
33	32	ralbii 2980	. . . 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 220	. . 3 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
35		ssrab2 3687	. . . . 5 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
36		fvex 6201	. . . . . . 7 ⊢ (𝑆‘𝑐) ∈ V
37		imaexg 7103	. . . . . . 7 ⊢ ((𝑆‘𝑐) ∈ V → ((𝑆‘𝑐) “ 𝑐) ∈ V)
38	36, 37	ax-mp 5	. . . . . 6 ⊢ ((𝑆‘𝑐) “ 𝑐) ∈ V
39	38, 1	dmmpti 6023	. . . . 5 ⊢ dom 𝑅 = (𝑂 ∖ 𝐸)
40	35, 39	sseqtr4i 3638	. . . 4 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ dom 𝑅
41		nfrab1 3122	. . . . 5 ⊢ Ⅎ𝑐{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
42		nfrab1 3122	. . . . 5 ⊢ Ⅎ𝑐{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
43		nfmpt1 4747	. . . . . 6 ⊢ Ⅎ𝑐(𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
44	1, 43	nfcxfr 2762	. . . . 5 ⊢ Ⅎ𝑐𝑅
45	41, 42, 44	funimass4f 29437	. . . 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 708	. . 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 221	. 2 ⊢ (𝑅 “ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
48		rabid 3116	. . . . . 6 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐))
49	14	adantr 481	. . . . . . 7 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) → (𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸))
50	3, 4, 5, 6, 7, 8, 9, 10	ballotlemic 30568	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) → (𝐼‘𝑐) ∈ 𝑐)
51	50	ex 450	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (¬ 1 ∈ 𝑐 → (𝐼‘𝑐) ∈ 𝑐))
52	51, 18	sylibrd 249	. . . . . . . 8 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (¬ 1 ∈ 𝑐 → 1 ∈ (𝑅‘𝑐)))
53	52	imp 445	. . . . . . 7 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) → 1 ∈ (𝑅‘𝑐))
54	49, 53	jca 554	. . . . . 6 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) → ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)))
55	48, 54	sylbi 207	. . . . 5 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} → ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)))
56	55	rgen 2922	. . . 4 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐))
57	25	elrab 3363	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑏} ↔ ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)))
58	28	cbvrabv 3199	. . . . . . 7 ⊢ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
59	58	eleq2i 2693	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑏} ↔ (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐})
60	57, 59	bitr3i 266	. . . . 5 ⊢ (((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)) ↔ (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐})
61	60	ralbii 2980	. . . 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 220	. . 3 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
63		ssrab2 3687	. . . . 5 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
64	63, 39	sseqtr4i 3638	. . . 4 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ dom 𝑅
65	42, 41, 44	funimass4f 29437	. . . 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 708	. . 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 221	. 2 ⊢ (𝑅 “ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
68	2, 12, 47, 67, 40, 64	rinvf1o 29432	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ifcif 4086  𝒫 cpw 4158   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114   ↾ cres 5116   “ cima 5117  Fun wfun 5882  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  infcinf 8347  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075   − cmin 10266   / cdiv 10684  ℕcn 11020  ℤcz 11377  ...cfz 12326  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-hash 13118

This theorem is referenced by:  ballotlem8  30598