Theorem vdn0frgrav2 26550

Description: A vertex in a friendship graph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 9-Dec-2017.)

Assertion

Ref	Expression
vdn0frgrav2	⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 0))

Proof of Theorem vdn0frgrav2

Dummy variables 𝑎 𝑏 𝑐 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frisusgra 26519	. . . . . . 7 ⊢ (𝑉 FriendGrph 𝐸 → 𝑉 USGrph 𝐸)
2	1	3ad2ant1 1075	. . . . . 6 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) → 𝑉 USGrph 𝐸)
3		simp3 1056	. . . . . 6 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ 𝑉)
4		simp2 1055	. . . . . 6 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) → 𝐸 ∈ Fin)
5	2, 3, 4	3jca 1235	. . . . 5 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉 ∧ 𝐸 ∈ Fin))
6	5	adantr 480	. . . 4 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → (𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉 ∧ 𝐸 ∈ Fin))
7		vdusgraval 26434	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}))
8	7	3adant3 1074	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ 𝑉 ∧ 𝐸 ∈ Fin) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}))
9	6, 8	syl 17	. . 3 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ((𝑉 VDeg 𝐸)‘𝑁) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}))
10		usgrafun 25878	. . . . . . . . 9 ⊢ (𝑉 USGrph 𝐸 → Fun 𝐸)
11		funfn 5833	. . . . . . . . 9 ⊢ (Fun 𝐸 ↔ 𝐸 Fn dom 𝐸)
12	10, 11	sylib 207	. . . . . . . 8 ⊢ (𝑉 USGrph 𝐸 → 𝐸 Fn dom 𝐸)
13	1, 12	syl 17	. . . . . . 7 ⊢ (𝑉 FriendGrph 𝐸 → 𝐸 Fn dom 𝐸)
14		3cyclfrgrarn 26540	. . . . . . . . . 10 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸))
15		preq1 4212	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑁 → {𝑎, 𝑏} = {𝑁, 𝑏})
16	15	eleq1d 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑁 → ({𝑎, 𝑏} ∈ ran 𝐸 ↔ {𝑁, 𝑏} ∈ ran 𝐸))
17		preq2 4213	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑁 → {𝑐, 𝑎} = {𝑐, 𝑁})
18	17	eleq1d 2672	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑁 → ({𝑐, 𝑎} ∈ ran 𝐸 ↔ {𝑐, 𝑁} ∈ ran 𝐸))
19	16, 18	3anbi13d 1393	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑁 → (({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)))
20	19	2rexbidv 3039	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑁 → (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) ↔ ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸)))
21	20	rspcva 3280	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ 𝑉 ∧ ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸))
22		fvelrnb 6153	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸 Fn dom 𝐸 → ({𝑁, 𝑏} ∈ ran 𝐸 ↔ ∃𝑥 ∈ dom 𝐸(𝐸‘𝑥) = {𝑁, 𝑏}))
23		prid1g 4239	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑁 ∈ 𝑉 → 𝑁 ∈ {𝑁, 𝑏})
24		eleq2 2677	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐸‘𝑥) = {𝑁, 𝑏} → (𝑁 ∈ (𝐸‘𝑥) ↔ 𝑁 ∈ {𝑁, 𝑏}))
25	23, 24	syl5ibrcom 236	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ 𝑉 → ((𝐸‘𝑥) = {𝑁, 𝑏} → 𝑁 ∈ (𝐸‘𝑥)))
26	25	reximdv 2999	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ 𝑉 → (∃𝑥 ∈ dom 𝐸(𝐸‘𝑥) = {𝑁, 𝑏} → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))
27	26	a1dd 48	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ 𝑉 → (∃𝑥 ∈ dom 𝐸(𝐸‘𝑥) = {𝑁, 𝑏} → (𝑏 ∈ 𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))))
28	27	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐸 ∈ Fin → (𝑁 ∈ 𝑉 → (∃𝑥 ∈ dom 𝐸(𝐸‘𝑥) = {𝑁, 𝑏} → (𝑏 ∈ 𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
29	28	com13 86	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃𝑥 ∈ dom 𝐸(𝐸‘𝑥) = {𝑁, 𝑏} → (𝑁 ∈ 𝑉 → (𝐸 ∈ Fin → (𝑏 ∈ 𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
30	22, 29	syl6bi 242	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸 Fn dom 𝐸 → ({𝑁, 𝑏} ∈ ran 𝐸 → (𝑁 ∈ 𝑉 → (𝐸 ∈ Fin → (𝑏 ∈ 𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))))))
31	30	com12 32	. . . . . . . . . . . . . . . . . . 19 ⊢ ({𝑁, 𝑏} ∈ ran 𝐸 → (𝐸 Fn dom 𝐸 → (𝑁 ∈ 𝑉 → (𝐸 ∈ Fin → (𝑏 ∈ 𝑉 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))))))
32	31	com25 97	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑁, 𝑏} ∈ ran 𝐸 → (𝑏 ∈ 𝑉 → (𝑁 ∈ 𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))))))
33	32	3ad2ant1 1075	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → (𝑏 ∈ 𝑉 → (𝑁 ∈ 𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))))))
34	33	com12 32	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ 𝑉 → (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → (𝑁 ∈ 𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))))))
35	34	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → (𝑁 ∈ 𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))))))
36	35	rexlimivv 3018	. . . . . . . . . . . . . 14 ⊢ (∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑁, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑁} ∈ ran 𝐸) → (𝑁 ∈ 𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
37	21, 36	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ 𝑉 ∧ ∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸)) → (𝑁 ∈ 𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
38	37	ex 449	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ 𝑉 → (∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → (𝑁 ∈ 𝑉 → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))))))
39	38	pm2.43a 52	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑉 → (∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → (𝐸 ∈ Fin → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
40	39	com3l 87	. . . . . . . . . 10 ⊢ (∀𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝑐} ∈ ran 𝐸 ∧ {𝑐, 𝑎} ∈ ran 𝐸) → (𝐸 ∈ Fin → (𝑁 ∈ 𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
41	14, 40	syl 17	. . . . . . . . 9 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉)) → (𝐸 ∈ Fin → (𝑁 ∈ 𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
42	41	expcom 450	. . . . . . . 8 ⊢ (1 < (#‘𝑉) → (𝑉 FriendGrph 𝐸 → (𝐸 ∈ Fin → (𝑁 ∈ 𝑉 → (𝐸 Fn dom 𝐸 → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))))))
43	42	com15 99	. . . . . . 7 ⊢ (𝐸 Fn dom 𝐸 → (𝑉 FriendGrph 𝐸 → (𝐸 ∈ Fin → (𝑁 ∈ 𝑉 → (1 < (#‘𝑉) → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))))))
44	13, 43	mpcom 37	. . . . . 6 ⊢ (𝑉 FriendGrph 𝐸 → (𝐸 ∈ Fin → (𝑁 ∈ 𝑉 → (1 < (#‘𝑉) → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥)))))
45	44	3imp1 1272	. . . . 5 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))
46		rexnal 2978	. . . . . 6 ⊢ (∃𝑥 ∈ dom 𝐸 ¬ ¬ 𝑁 ∈ (𝐸‘𝑥) ↔ ¬ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸‘𝑥))
47		notnotb 303	. . . . . . . 8 ⊢ (𝑁 ∈ (𝐸‘𝑥) ↔ ¬ ¬ 𝑁 ∈ (𝐸‘𝑥))
48	47	bicomi 213	. . . . . . 7 ⊢ (¬ ¬ 𝑁 ∈ (𝐸‘𝑥) ↔ 𝑁 ∈ (𝐸‘𝑥))
49	48	rexbii 3023	. . . . . 6 ⊢ (∃𝑥 ∈ dom 𝐸 ¬ ¬ 𝑁 ∈ (𝐸‘𝑥) ↔ ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))
50	46, 49	bitr3i 265	. . . . 5 ⊢ (¬ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸‘𝑥) ↔ ∃𝑥 ∈ dom 𝐸 𝑁 ∈ (𝐸‘𝑥))
51	45, 50	sylibr 223	. . . 4 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ¬ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸‘𝑥))
52		dmfi 8129	. . . . . . . . 9 ⊢ (𝐸 ∈ Fin → dom 𝐸 ∈ Fin)
53	52	3ad2ant2 1076	. . . . . . . 8 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) → dom 𝐸 ∈ Fin)
54	53	adantr 480	. . . . . . 7 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → dom 𝐸 ∈ Fin)
55		rabexg 4739	. . . . . . 7 ⊢ (dom 𝐸 ∈ Fin → {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∈ V)
56		hasheq0 13015	. . . . . . 7 ⊢ ({𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} ∈ V → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) = 0 ↔ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} = ∅))
57	54, 55, 56	3syl 18	. . . . . 6 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) = 0 ↔ {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} = ∅))
58		rabeq0 3911	. . . . . 6 ⊢ ({𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)} = ∅ ↔ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸‘𝑥))
59	57, 58	syl6bb 275	. . . . 5 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) = 0 ↔ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸‘𝑥)))
60	59	necon3abid 2818	. . . 4 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ((#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) ≠ 0 ↔ ¬ ∀𝑥 ∈ dom 𝐸 ¬ 𝑁 ∈ (𝐸‘𝑥)))
61	51, 60	mpbird 246	. . 3 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}) ≠ 0)
62	9, 61	eqnetrd 2849	. 2 ⊢ (((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) ∧ 1 < (#‘𝑉)) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 0)
63	62	ex 449	1 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝐸 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (1 < (#‘𝑉) → ((𝑉 VDeg 𝐸)‘𝑁) ≠ 0))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ∅c0 3874  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  1c1 9816   < clt 9953  #chash 12979   USGrph cusg 25859   VDeg cvdg 26420   FriendGrph cfrgra 26515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-usgra 25862  df-vdgr 26421  df-frgra 26516

This theorem is referenced by: (None)