Theorem friendshipgt3 30478

Description: The friendship theorem for big graphs: In every finite friendship graph with order greater than 3 there is a vertex which is adjacent to all other vertices. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)

Hypothesis

Ref	Expression
frgrreggt1.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
friendshipgt3	⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))

Distinct variable groups:   𝑣,𝐺   𝑣,𝑉   𝑤,𝐺,𝑣   𝑤,𝑉

Proof of Theorem friendshipgt3

Dummy variables 𝑘 𝑚 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrreggt1.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2737	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	frgrregorufrg 30406	. . 3 ⊢ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
4	3	3ad2ant1 1134	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∀𝑘 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
5	1	frgrogt3nreg 30477	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘)
6		frgrusgr 30341	. . . . . . 7 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
7	6	anim1i 616	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
8	1	isfusgr 29396	. . . . . 6 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
9	7, 8	sylibr 234	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
10	9	3adant3 1133	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 𝐺 ∈ FinUSGraph)
11		0red 11140	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 0 ∈ ℝ)
12		3re 12230	. . . . . . . . 9 ⊢ 3 ∈ ℝ
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 3 ∈ ℝ)
14		hashcl 14284	. . . . . . . . . 10 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
15	14	nn0red 12468	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℝ)
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → (♯‘𝑉) ∈ ℝ)
17		3pos 12255	. . . . . . . . 9 ⊢ 0 < 3
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 0 < 3)
19		simpr 484	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 3 < (♯‘𝑉))
20	11, 13, 16, 18, 19	lttrd 11299	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 0 < (♯‘𝑉))
21	20	gt0ne0d 11706	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → (♯‘𝑉) ≠ 0)
22		hasheq0 14291	. . . . . . . 8 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅))
23	22	adantr 480	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅))
24	23	necon3bid 2977	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ((♯‘𝑉) ≠ 0 ↔ 𝑉 ≠ ∅))
25	21, 24	mpbid 232	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 𝑉 ≠ ∅)
26	25	3adant1 1131	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 𝑉 ≠ ∅)
27	1	fusgrn0degnn0 29578	. . . 4 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → ∃𝑡 ∈ 𝑉 ∃𝑚 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑡) = 𝑚)
28	10, 26, 27	syl2anc 585	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∃𝑡 ∈ 𝑉 ∃𝑚 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑡) = 𝑚)
29		r19.26 3097	. . . . . . . 8 ⊢ (∀𝑘 ∈ ℕ0 ((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) ↔ (∀𝑘 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘))
30		simpllr 776	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) → 𝑚 ∈ ℕ0)
31		fveqeq2 6844	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑡 → (((VtxDeg‘𝐺)‘𝑢) = 𝑚 ↔ ((VtxDeg‘𝐺)‘𝑡) = 𝑚))
32	31	rspcev 3577	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝑉 ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) → ∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚)
33	32	ad4ant13 752	. . . . . . . . . . . . 13 ⊢ ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) → ∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚)
34		ornld 1062	. . . . . . . . . . . . 13 ⊢ (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑚) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
35	33, 34	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) → (((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑚) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
36	35	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) ∧ 𝑘 = 𝑚) → (((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑚) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
37		eqeq2 2749	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (((VtxDeg‘𝐺)‘𝑢) = 𝑘 ↔ ((VtxDeg‘𝐺)‘𝑢) = 𝑚))
38	37	rexbidv 3161	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 ↔ ∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚))
39		breq2 5103	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝐺 RegUSGraph 𝑘 ↔ 𝐺 RegUSGraph 𝑚))
40	39	orbi1d 917	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → ((𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) ↔ (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
41	38, 40	imbi12d 344	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → ((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ↔ (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))))
42	39	notbid 318	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (¬ 𝐺 RegUSGraph 𝑘 ↔ ¬ 𝐺 RegUSGraph 𝑚))
43	41, 42	anbi12d 633	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → (((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) ↔ ((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑚)))
44	43	imbi1d 341	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → ((((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) ↔ (((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑚) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
45	44	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) ∧ 𝑘 = 𝑚) → ((((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) ↔ (((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑚) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
46	36, 45	mpbird 257	. . . . . . . . . 10 ⊢ (((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) ∧ 𝑘 = 𝑚) → (((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
47	30, 46	rspcimdv 3567	. . . . . . . . 9 ⊢ ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) → (∀𝑘 ∈ ℕ0 ((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
48	47	com12 32	. . . . . . . 8 ⊢ (∀𝑘 ∈ ℕ0 ((∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) → ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
49	29, 48	sylbir 235	. . . . . . 7 ⊢ ((∀𝑘 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘) → ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
50	49	expcom 413	. . . . . 6 ⊢ (∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘 → (∀𝑘 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) → ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
51	50	com13 88	. . . . 5 ⊢ ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) → (∀𝑘 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) → (∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
52	51	exp31 419	. . . 4 ⊢ ((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) → (((VtxDeg‘𝐺)‘𝑡) = 𝑚 → ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → (∀𝑘 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) → (∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))))
53	52	rexlimivv 3179	. . 3 ⊢ (∃𝑡 ∈ 𝑉 ∃𝑚 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑡) = 𝑚 → ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → (∀𝑘 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) → (∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))))
54	28, 53	mpcom 38	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → (∀𝑘 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) → (∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
55	4, 5, 54	mp2d 49	1 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3061   ∖ cdif 3899  ∅c0 4286  {csn 4581  {cpr 4583   class class class wbr 5099  ‘cfv 6493  Fincfn 8888  ℝcr 11030  0cc0 11031   < clt 11171  3c3 12206  ℕ₀cn0 12406  ♯chash 14258  Vtxcvtx 29074  Edgcedg 29125  USGraphcusgr 29227  FinUSGraphcfusgr 29394  VtxDegcvtxdg 29544   RegUSGraph crusgr 29635   FriendGraph cfrgr 30338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7683  ax-inf2 9555  ax-cnex 11087  ax-resscn 11088  ax-1cn 11089  ax-icn 11090  ax-addcl 11091  ax-addrcl 11092  ax-mulcl 11093  ax-mulrcl 11094  ax-mulcom 11095  ax-addass 11096  ax-mulass 11097  ax-distr 11098  ax-i2m1 11099  ax-1ne0 11100  ax-1rid 11101  ax-rnegex 11102  ax-rrecex 11103  ax-cnre 11104  ax-pre-lttri 11105  ax-pre-lttrn 11106  ax-pre-ltadd 11107  ax-pre-mulgt0 11108  ax-pre-sup 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-disj 5067  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-oadd 8404  df-er 8638  df-ec 8640  df-qs 8644  df-map 8770  df-pm 8771  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-inf 9351  df-oi 9420  df-dju 9818  df-card 9856  df-pnf 11173  df-mnf 11174  df-xr 11175  df-ltxr 11176  df-le 11177  df-sub 11371  df-neg 11372  df-div 11800  df-nn 12151  df-2 12213  df-3 12214  df-n0 12407  df-xnn0 12480  df-z 12494  df-uz 12757  df-rp 12911  df-xadd 13032  df-ico 13272  df-fz 13429  df-fzo 13576  df-fl 13717  df-mod 13795  df-seq 13930  df-exp 13990  df-hash 14259  df-word 14442  df-lsw 14491  df-concat 14499  df-s1 14525  df-substr 14570  df-pfx 14600  df-reps 14697  df-csh 14717  df-s2 14776  df-s3 14777  df-cj 15027  df-re 15028  df-im 15029  df-sqrt 15163  df-abs 15164  df-clim 15416  df-sum 15615  df-dvds 16185  df-gcd 16427  df-prm 16604  df-phi 16698  df-vtx 29076  df-iedg 29077  df-edg 29126  df-uhgr 29136  df-ushgr 29137  df-upgr 29160  df-umgr 29161  df-uspgr 29228  df-usgr 29229  df-fusgr 29395  df-nbgr 29411  df-vtxdg 29545  df-rgr 29636  df-rusgr 29637  df-wlks 29678  df-wlkson 29679  df-trls 29769  df-trlson 29770  df-pths 29792  df-spths 29793  df-pthson 29794  df-spthson 29795  df-wwlks 29908  df-wwlksn 29909  df-wwlksnon 29910  df-wspthsn 29911  df-wspthsnon 29912  df-clwwlk 30062  df-clwwlkn 30105  df-clwwlknon 30168  df-conngr 30267  df-frgr 30339

This theorem is referenced by:  friendship  30479