Theorem friendshipgt3 28179

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 2823	. . . 4 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	frgrregorufrg 28107	. . 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 1129	. 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 28178	. 2 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘)
6		frgrusgr 28042	. . . . . . 7 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
7	6	anim1i 616	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
8	1	isfusgr 27102	. . . . . 6 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
9	7, 8	sylibr 236	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
10	9	3adant3 1128	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 𝐺 ∈ FinUSGraph)
11		0red 10646	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 0 ∈ ℝ)
12		3re 11720	. . . . . . . . 9 ⊢ 3 ∈ ℝ
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 3 ∈ ℝ)
14		hashcl 13720	. . . . . . . . . 10 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℕ0)
15	14	nn0red 11959	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → (♯‘𝑉) ∈ ℝ)
16	15	adantr 483	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → (♯‘𝑉) ∈ ℝ)
17		3pos 11745	. . . . . . . . 9 ⊢ 0 < 3
18	17	a1i 11	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 0 < 3)
19		simpr 487	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 3 < (♯‘𝑉))
20	11, 13, 16, 18, 19	lttrd 10803	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 0 < (♯‘𝑉))
21	20	gt0ne0d 11206	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → (♯‘𝑉) ≠ 0)
22		hasheq0 13727	. . . . . . . 8 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅))
23	22	adantr 483	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅))
24	23	necon3bid 3062	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ((♯‘𝑉) ≠ 0 ↔ 𝑉 ≠ ∅))
25	21, 24	mpbid 234	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 𝑉 ≠ ∅)
26	25	3adant1 1126	. . . 4 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → 𝑉 ≠ ∅)
27	1	fusgrn0degnn0 27283	. . . 4 ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → ∃𝑡 ∈ 𝑉 ∃𝑚 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑡) = 𝑚)
28	10, 26, 27	syl2anc 586	. . 3 ⊢ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∃𝑡 ∈ 𝑉 ∃𝑚 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑡) = 𝑚)
29		r19.26 3172	. . . . . . . 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 774	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) → 𝑚 ∈ ℕ0)
31		fveqeq2 6681	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑡 → (((VtxDeg‘𝐺)‘𝑢) = 𝑚 ↔ ((VtxDeg‘𝐺)‘𝑡) = 𝑚))
32	31	rspcev 3625	. . . . . . . . . . . . . 14 ⊢ ((𝑡 ∈ 𝑉 ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) → ∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚)
33	32	ad4ant13 749	. . . . . . . . . . . . 13 ⊢ ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉))) → ∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚)
34		ornld 1056	. . . . . . . . . . . . 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 483	. . . . . . . . . . 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 2835	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (((VtxDeg‘𝐺)‘𝑢) = 𝑘 ↔ ((VtxDeg‘𝐺)‘𝑢) = 𝑚))
38	37	rexbidv 3299	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 ↔ ∃𝑢 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚))
39		breq2 5072	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝐺 RegUSGraph 𝑘 ↔ 𝐺 RegUSGraph 𝑚))
40	39	orbi1d 913	. . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . 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 320	. . . . . . . . . . . . . 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 344	. . . . . . . . . . . 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 484	. . . . . . . . . . 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 259	. . . . . . . . . 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 3615	. . . . . . . . 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 237	. . . . . . 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 416	. . . . . 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 422	. . . 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 3294	. . 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 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  ∃wrex 3141   ∖ cdif 3935  ∅c0 4293  {csn 4569  {cpr 4571   class class class wbr 5068  ‘cfv 6357  Fincfn 8511  ℝcr 10538  0cc0 10539   < clt 10677  3c3 11696  ℕ₀cn0 11900  ♯chash 13693  Vtxcvtx 26783  Edgcedg 26834  USGraphcusgr 26936  FinUSGraphcfusgr 27100  VtxDegcvtxdg 27249   RegUSGraph crusgr 27340   FriendGraph cfrgr 28039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-ac2 9887  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-disj 5034  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-ec 8293  df-qs 8297  df-map 8410  df-pm 8411  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-sup 8908  df-inf 8909  df-oi 8976  df-dju 9332  df-card 9370  df-ac 9544  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-xnn0 11971  df-z 11985  df-uz 12247  df-rp 12393  df-xadd 12511  df-ico 12747  df-fz 12896  df-fzo 13037  df-fl 13165  df-mod 13241  df-seq 13373  df-exp 13433  df-hash 13694  df-word 13865  df-lsw 13917  df-concat 13925  df-s1 13952  df-substr 14005  df-pfx 14035  df-reps 14133  df-csh 14153  df-s2 14212  df-s3 14213  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-sum 15045  df-dvds 15610  df-gcd 15846  df-prm 16018  df-phi 16105  df-vtx 26785  df-iedg 26786  df-edg 26835  df-uhgr 26845  df-ushgr 26846  df-upgr 26869  df-umgr 26870  df-uspgr 26937  df-usgr 26938  df-fusgr 27101  df-nbgr 27117  df-vtxdg 27250  df-rgr 27341  df-rusgr 27342  df-wlks 27383  df-wlkson 27384  df-trls 27476  df-trlson 27477  df-pths 27499  df-spths 27500  df-pthson 27501  df-spthson 27502  df-wwlks 27610  df-wwlksn 27611  df-wwlksnon 27612  df-wspthsn 27613  df-wspthsnon 27614  df-clwwlk 27762  df-clwwlkn 27805  df-clwwlknon 27869  df-conngr 27968  df-frgr 28040

This theorem is referenced by:  friendship  28180