Theorem friendshipgt3 30379

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 2729	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	frgrregorufrg 30307	. . 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 1133	. 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 30378	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘)
6		frgrusgr 30242	. . . . . . 7 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
7	6	anim1i 615	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
8	1	isfusgr 29300	. . . . . 6 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
9	7, 8	sylibr 234	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
10	9	3adant3 1132	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 𝐺 ∈ FinUSGraph)
11		0red 11156	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 0 ∈ ℝ)
12		3re 12245	. . . . . . . . 9 ⊢ 3 ∈ ℝ
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 3 ∈ ℝ)
14		hashcl 14300	. . . . . . . . . 10 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
15	14	nn0red 12483	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℝ)
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → (♯‘𝑉) ∈ ℝ)
17		3pos 12270	. . . . . . . . 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 11314	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 0 < (♯‘𝑉))
21	20	gt0ne0d 11721	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → (♯‘𝑉) ≠ 0)
22		hasheq0 14307	. . . . . . . 8 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅))
23	22	adantr 480	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅))
24	23	necon3bid 2969	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ((♯‘𝑉) ≠ 0 ↔ 𝑉 ≠ ∅))
25	21, 24	mpbid 232	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 𝑉 ≠ ∅)
26	25	3adant1 1130	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 𝑉 ≠ ∅)
27	1	fusgrn0degnn0 29482	. . . 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 3091	. . . . . . . 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 775	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) → 𝑚 ∈ ℕ0)
31		fveqeq2 6850	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑡 → (((VtxDeg‘𝐺)‘𝑢) = 𝑚 ↔ ((VtxDeg‘𝐺)‘𝑡) = 𝑚))
32	31	rspcev 3585	. . . . . . . . . . . . . 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 1061	. . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (((VtxDeg‘𝐺)‘𝑢) = 𝑘 ↔ ((VtxDeg‘𝐺)‘𝑢) = 𝑚))
38	37	rexbidv 3157	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 ↔ ∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚))
39		breq2 5106	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝐺 RegUSGraph 𝑘 ↔ 𝐺 RegUSGraph 𝑚))
40	39	orbi1d 916	. . . . . . . . . . . . . . 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 3575	. . . . . . . . 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 3177	. . 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 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ∖ cdif 3908  ∅c0 4292  {csn 4585  {cpr 4587   class class class wbr 5102  ‘cfv 6500  Fincfn 8896  ℝcr 11046  0cc0 11047   < clt 11187  3c3 12221  ℕ₀cn0 12421  ♯chash 14274  Vtxcvtx 28978  Edgcedg 29029  USGraphcusgr 29131  FinUSGraphcfusgr 29298  VtxDegcvtxdg 29448   RegUSGraph crusgr 29539   FriendGraph cfrgr 30239

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7692  ax-inf2 9573  ax-ac2 10395  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124  ax-pre-sup 11125

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-disj 5070  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6263  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6453  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7824  df-1st 7948  df-2nd 7949  df-frecs 8238  df-wrecs 8269  df-recs 8318  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8649  df-ec 8651  df-qs 8655  df-map 8779  df-pm 8780  df-en 8897  df-dom 8898  df-sdom 8899  df-fin 8900  df-sup 9370  df-inf 9371  df-oi 9440  df-dju 9833  df-card 9871  df-ac 10048  df-pnf 11189  df-mnf 11190  df-xr 11191  df-ltxr 11192  df-le 11193  df-sub 11386  df-neg 11387  df-div 11815  df-nn 12166  df-2 12228  df-3 12229  df-n0 12422  df-xnn0 12495  df-z 12509  df-uz 12773  df-rp 12931  df-xadd 13052  df-ico 13291  df-fz 13448  df-fzo 13595  df-fl 13733  df-mod 13811  df-seq 13946  df-exp 14006  df-hash 14275  df-word 14458  df-lsw 14507  df-concat 14515  df-s1 14540  df-substr 14585  df-pfx 14615  df-reps 14712  df-csh 14732  df-s2 14792  df-s3 14793  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15432  df-sum 15631  df-dvds 16201  df-gcd 16443  df-prm 16620  df-phi 16714  df-vtx 28980  df-iedg 28981  df-edg 29030  df-uhgr 29040  df-ushgr 29041  df-upgr 29064  df-umgr 29065  df-uspgr 29132  df-usgr 29133  df-fusgr 29299  df-nbgr 29315  df-vtxdg 29449  df-rgr 29540  df-rusgr 29541  df-wlks 29582  df-wlkson 29583  df-trls 29673  df-trlson 29674  df-pths 29696  df-spths 29697  df-pthson 29698  df-spthson 29699  df-wwlks 29812  df-wwlksn 29813  df-wwlksnon 29814  df-wspthsn 29815  df-wspthsnon 29816  df-clwwlk 29963  df-clwwlkn 30006  df-clwwlknon 30069  df-conngr 30168  df-frgr 30240

This theorem is referenced by:  friendship  30380