Theorem friendshipgt3 30417

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 30345	. . 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 30416	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘)
6		frgrusgr 30280	. . . . . . 7 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
7	6	anim1i 615	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
8	1	isfusgr 29335	. . . . . 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 11264	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 0 ∈ ℝ)
12		3re 12346	. . . . . . . . 9 ⊢ 3 ∈ ℝ
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 3 ∈ ℝ)
14		hashcl 14395	. . . . . . . . . 10 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
15	14	nn0red 12588	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℝ)
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → (♯‘𝑉) ∈ ℝ)
17		3pos 12371	. . . . . . . . 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 11422	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 0 < (♯‘𝑉))
21	20	gt0ne0d 11827	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → (♯‘𝑉) ≠ 0)
22		hasheq0 14402	. . . . . . . 8 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅))
23	22	adantr 480	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅))
24	23	necon3bid 2985	. . . . . 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 29517	. . . 4 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → ∃𝑡 ∈ 𝑉 ∃𝑚 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑡) = 𝑚)
28	10, 26, 27	syl2anc 584	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∃𝑡 ∈ 𝑉 ∃𝑚 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑡) = 𝑚)
29		r19.26 3111	. . . . . . . 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 6915	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑡 → (((VtxDeg‘𝐺)‘𝑢) = 𝑚 ↔ ((VtxDeg‘𝐺)‘𝑡) = 𝑚))
32	31	rspcev 3622	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝑉 ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) → ∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚)
33	32	ad4ant13 751	. . . . . . . . . . . . 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 3179	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 ↔ ∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚))
39		breq2 5147	. . . . . . . . . . . . . . . 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 632	. . . . . . . . . . . . 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 3612	. . . . . . . . 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 3201	. . 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 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948  ∅c0 4333  {csn 4626  {cpr 4628   class class class wbr 5143  ‘cfv 6561  Fincfn 8985  ℝcr 11154  0cc0 11155   < clt 11295  3c3 12322  ℕ₀cn0 12526  ♯chash 14369  Vtxcvtx 29013  Edgcedg 29064  USGraphcusgr 29166  FinUSGraphcfusgr 29333  VtxDegcvtxdg 29483   RegUSGraph crusgr 29574   FriendGraph cfrgr 30277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-xadd 13155  df-ico 13393  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-reps 14807  df-csh 14827  df-s2 14887  df-s3 14888  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-dvds 16291  df-gcd 16532  df-prm 16709  df-phi 16803  df-vtx 29015  df-iedg 29016  df-edg 29065  df-uhgr 29075  df-ushgr 29076  df-upgr 29099  df-umgr 29100  df-uspgr 29167  df-usgr 29168  df-fusgr 29334  df-nbgr 29350  df-vtxdg 29484  df-rgr 29575  df-rusgr 29576  df-wlks 29617  df-wlkson 29618  df-trls 29710  df-trlson 29711  df-pths 29734  df-spths 29735  df-pthson 29736  df-spthson 29737  df-wwlks 29850  df-wwlksn 29851  df-wwlksnon 29852  df-wspthsn 29853  df-wspthsnon 29854  df-clwwlk 30001  df-clwwlkn 30044  df-clwwlknon 30107  df-conngr 30206  df-frgr 30278

This theorem is referenced by:  friendship  30418