Theorem ballotlem7 35221

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 6587	. 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 35219	. 2 ⊢ ◡𝑅 = 𝑅
13		rabid 3435	. . . . . 6 ⊢ (𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ↔ (𝑐 ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ 𝑐))
14	3, 4, 5, 6, 7, 8, 9, 10, 11, 1	ballotlemrc 35216	. . . . . . . 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 35193	. . . . . . . . . 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 35220	. . . . . . . . . 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 3081	. . . 4 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐))
25		eleq2 2852	. . . . . . . 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 3647	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑏} ↔ ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ ¬ 1 ∈ (𝑅‘𝑐)))
28		eleq2 2852	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (1 ∈ 𝑏 ↔ 1 ∈ 𝑐))
29	28	notbid 321	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (¬ 1 ∈ 𝑏 ↔ ¬ 1 ∈ 𝑐))
30	29	cbvrabv 3425	. . . . . . 7 ⊢ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
31	30	eleq2i 2855	. . . . . 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 3111	. . . 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 4007	. . . . 5 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
36		fvex 6906	. . . . . . 7 ⊢ (𝑆‘𝑐) ∈ V
37		imaexg 7932	. . . . . . 7 ⊢ ((𝑆‘𝑐) ∈ V → ((𝑆‘𝑐) “ 𝑐) ∈ V)
38	36, 37	ax-mp 5	. . . . . 6 ⊢ ((𝑆‘𝑐) “ 𝑐) ∈ V
39	38, 1	dmmpti 6691	. . . . 5 ⊢ dom 𝑅 = (𝑂 ∖ 𝐸)
40	35, 39	sseqtrri 3961	. . . 4 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐} ⊆ dom 𝑅
41		nfrab1 3434	. . . . 5 ⊢ Ⅎ𝑐{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
42		nfrab1 3434	. . . . 5 ⊢ Ⅎ𝑐{𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐}
43		nfmpt1 5206	. . . . . 6 ⊢ Ⅎ𝑐(𝑐 ∈ (𝑂 ∖ 𝐸) ↦ ((𝑆‘𝑐) “ 𝑐))
44	1, 43	nfcxfr 2923	. . . . 5 ⊢ Ⅎ𝑐𝑅
45	41, 42, 44	funimass4f 33285	. . . 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 3435	. . . . . 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 35192	. . . . . . . . . 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 3081	. . . 4 ⊢ ∀𝑐 ∈ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐))
57	25	elrab 3647	. . . . . 6 ⊢ ((𝑅‘𝑐) ∈ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑏} ↔ ((𝑅‘𝑐) ∈ (𝑂 ∖ 𝐸) ∧ 1 ∈ (𝑅‘𝑐)))
58	28	cbvrabv 3425	. . . . . . 7 ⊢ {𝑏 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑏} = {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ 1 ∈ 𝑐}
59	58	eleq2i 2855	. . . . . 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 3111	. . . 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 4007	. . . . 5 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ (𝑂 ∖ 𝐸)
64	63, 39	sseqtrri 3961	. . . 4 ⊢ {𝑐 ∈ (𝑂 ∖ 𝐸) ∣ ¬ 1 ∈ 𝑐} ⊆ dom 𝑅
65	42, 41, 44	funimass4f 33285	. . . 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 33278	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 1569   ∈ wcel 2146  ∀wral 3079  {crab 3415  Vcvv 3453   ⊆ wss 3899   ∖ cdif 4047   ∩ cin 4091  ifcif 4484  𝒫 cpw 4559   class class class wbr 5105   ↦ cmpt 5188  dom cdm 5654   ↾ cres 5656   “ cima 5657  Fun wfun 6540  –1-1-onto→wf1o 6545  ‘cfv 6546  (class class class)co 7427  infcinf 9443  ℝcr 11209  0cc0 11210  1c1 11211   + caddc 11213   < clt 11353   ≤ cle 11354   − cmin 11551   / cdiv 11983  ℕcn 12345  ℤcz 12703  ...cfz 13649  ♯chash 14484

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-1o 8487  df-oadd 8491  df-er 8728  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9444  df-inf 9445  df-dju 9992  df-card 10030  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-nn 12346  df-2 12415  df-n0 12617  df-z 12704  df-uz 12976  df-rp 13131  df-fz 13650  df-hash 14485

This theorem is used by:  ballotlem8  35222