Theorem ballotlem7 34993

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 6579	. 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 34991	. 2 ⊢ ◡𝑅 = 𝑅
13		rabid 3439	. . . . . 6 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐))
14	3, 4, 5, 6, 7, 8, 9, 10, 11, 1	ballotlemrc 34988	. . . . . . . 8 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸))
15	14	adantr 486	. . . . . . 7 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐) → (𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸))
16	3, 4, 5, 6, 7, 8, 9, 10	ballotlem1c 34965	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐) → ¬ (𝐼‘𝑐) ∈ 𝑐)
17	16	ex 418	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (1 ∈ 𝑐 → ¬ (𝐼‘𝑐) ∈ 𝑐))
18	3, 4, 5, 6, 7, 8, 9, 10, 11, 1	ballotlem1ri 34992	. . . . . . . . . 10 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (1 ∈ (𝑅‘𝑐) ↔ (𝐼‘𝑐) ∈ 𝑐))
19	18	notbid 321	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (¬ 1 ∈ (𝑅‘𝑐) ↔ ¬ (𝐼‘𝑐) ∈ 𝑐))
20	17, 19	sylibrd 262	. . . . . . . 8 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (1 ∈ 𝑐 → ¬ 1 ∈ (𝑅‘𝑐)))
21	20	imp 412	. . . . . . 7 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐) → ¬ 1 ∈ (𝑅‘𝑐))
22	15, 21	jca 521	. . . . . 6 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐) → ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)))
23	13, 22	sylbi 220	. . . . 5 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} → ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)))
24	23	rgen 3083	. . . 4 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐))
25		eleq2 2854	. . . . . . . 8 ⊢ (𝑏 = (𝑅‘𝑐) → (1 ∈ 𝑏 ↔ 1 ∈ (𝑅‘𝑐)))
26	25	notbid 321	. . . . . . 7 ⊢ (𝑏 = (𝑅‘𝑐) → (¬ 1 ∈ 𝑏 ↔ ¬ 1 ∈ (𝑅‘𝑐)))
27	26	elrab 3652	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑏} ↔ ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)))
28		eleq2 2854	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (1 ∈ 𝑏 ↔ 1 ∈ 𝑐))
29	28	notbid 321	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (¬ 1 ∈ 𝑏 ↔ ¬ 1 ∈ 𝑐))
30	29	cbvrabv 3428	. . . . . . 7 ⊢ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
31	30	eleq2i 2857	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑏} ↔ (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
32	27, 31	bitr3i 280	. . . . 5 ⊢ (((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)) ↔ (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐})
33	32	ralbii 3113	. . . 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 233	. . 3 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
35		ssrab2 4035	. . . . 5 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
36		fvex 6898	. . . . . . 7 ⊢ (𝑆‘𝑐) ∈ V
37		imaexg 7916	. . . . . . 7 ⊢ ((𝑆‘𝑐) ∈ V → ((𝑆‘𝑐) “ 𝑐) ∈ V)
38	36, 37	ax-mp 5	. . . . . 6 ⊢ ((𝑆‘𝑐) “ 𝑐) ∈ V
39	38, 1	dmmpti 6683	. . . . 5 ⊢ dom 𝑅 = (𝑂 ∖ 𝐸)
40	35, 39	sseqtrri 3987	. . . 4 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ dom 𝑅
41		nfrab1 3438	. . . . 5 ⊢ Ⅎ𝑐{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
42		nfrab1 3438	. . . . 5 ⊢ Ⅎ𝑐{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
43		nfmpt1 5212	. . . . . 6 ⊢ Ⅎ𝑐(𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
44	1, 43	nfcxfr 2925	. . . . 5 ⊢ Ⅎ𝑐𝑅
45	41, 42, 44	funimass4f 33055	. . . 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 705	. . 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 234	. 2 ⊢ (𝑅 “ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
48		rabid 3439	. . . . . 6 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐))
49	14	adantr 486	. . . . . . 7 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) → (𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸))
50	3, 4, 5, 6, 7, 8, 9, 10	ballotlemic 34964	. . . . . . . . . 10 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) → (𝐼‘𝑐) ∈ 𝑐)
51	50	ex 418	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (¬ 1 ∈ 𝑐 → (𝐼‘𝑐) ∈ 𝑐))
52	51, 18	sylibrd 262	. . . . . . . 8 ⊢ (𝑐 ∈ (𝑂 ∖ 𝐸) → (¬ 1 ∈ 𝑐 → 1 ∈ (𝑅‘𝑐)))
53	52	imp 412	. . . . . . 7 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) → 1 ∈ (𝑅‘𝑐))
54	49, 53	jca 521	. . . . . 6 ⊢ ((𝑐 ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ 𝑐) → ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)))
55	48, 54	sylbi 220	. . . . 5 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} → ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)))
56	55	rgen 3083	. . . 4 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐))
57	25	elrab 3652	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑏} ↔ ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)))
58	28	cbvrabv 3428	. . . . . . 7 ⊢ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
59	58	eleq2i 2857	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑏} ↔ (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐})
60	57, 59	bitr3i 280	. . . . 5 ⊢ (((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)) ↔ (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐})
61	60	ralbii 3113	. . . 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 233	. . 3 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} (𝑅‘𝑐) ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
63		ssrab2 4035	. . . . 5 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
64	63, 39	sseqtrri 3987	. . . 4 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ dom 𝑅
65	42, 41, 44	funimass4f 33055	. . . 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 705	. . 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 234	. 2 ⊢ (𝑅 “ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}) ⊆ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
68	2, 12, 47, 67, 40, 64	rinvf1o 33048	1 ⊢ (𝑅 ↾ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}):{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}–1-1-onto→{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418  Vcvv 3457   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906  ifcif 4489  𝒫 cpw 4564   class class class wbr 5111   ↦ cmpt 5194  dom cdm 5663   ↾ cres 5665   “ cima 5666  Fun wfun 6534  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7419  infcinf 9408  ℝcr 11116  0cc0 11117  1c1 11118   + caddc 11120   < clt 11260   ≤ cle 11261   − cmin 11458   / cdiv 11888  ℕcn 12250  ℤcz 12608  ...cfz 13553  ♯chash 14386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-sup 9409  df-inf 9410  df-dju 9903  df-card 9941  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-nn 12251  df-2 12320  df-n0 12522  df-z 12609  df-uz 12881  df-rp 13035  df-fz 13554  df-hash 14387

This theorem is used by:  ballotlem8  34994