Theorem friendshipgt3 26648

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.)

Assertion

Ref	Expression
friendshipgt3	⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)

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

Proof of Theorem friendshipgt3

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

Step	Hyp	Ref	Expression
1		frgraregorufrg 26599	. . 3 ⊢ (𝑉 FriendGrph 𝐸 → ∀𝑛 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
2	1	3ad2ant1 1075	. 2 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∀𝑛 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
3		frgraogt3nreg 26647	. 2 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∀𝑛 ∈ ℕ0 ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛)
4		frisusgra 26519	. . . . 5 ⊢ (𝑉 FriendGrph 𝐸 → 𝑉 USGrph 𝐸)
5	4	3ad2ant1 1075	. . . 4 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 𝑉 USGrph 𝐸)
6		simp2 1055	. . . 4 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 𝑉 ∈ Fin)
7		0red 9920	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 0 ∈ ℝ)
8		3re 10971	. . . . . . . . 9 ⊢ 3 ∈ ℝ
9	8	a1i 11	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 3 ∈ ℝ)
10		hashcl 13009	. . . . . . . . . 10 ⊢ (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
11	10	nn0red 11229	. . . . . . . . 9 ⊢ (𝑉 ∈ Fin → (#‘𝑉) ∈ ℝ)
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → (#‘𝑉) ∈ ℝ)
13		3pos 10991	. . . . . . . . 9 ⊢ 0 < 3
14	13	a1i 11	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 0 < 3)
15		simpr 476	. . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 3 < (#‘𝑉))
16	7, 9, 12, 14, 15	lttrd 10077	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 0 < (#‘𝑉))
17	16	gt0ne0d 10471	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → (#‘𝑉) ≠ 0)
18		hasheq0 13015	. . . . . . . 8 ⊢ (𝑉 ∈ Fin → ((#‘𝑉) = 0 ↔ 𝑉 = ∅))
19	18	adantr 480	. . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ((#‘𝑉) = 0 ↔ 𝑉 = ∅))
20	19	necon3bid 2826	. . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ((#‘𝑉) ≠ 0 ↔ 𝑉 ≠ ∅))
21	17, 20	mpbid 221	. . . . 5 ⊢ ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 𝑉 ≠ ∅)
22	21	3adant1 1072	. . . 4 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 𝑉 ≠ ∅)
23		usgn0fidegnn0 26556	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ∃𝑡 ∈ 𝑉 ∃𝑚 ∈ ℕ0 ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚)
24	5, 6, 22, 23	syl3anc 1318	. . 3 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∃𝑡 ∈ 𝑉 ∃𝑚 ∈ ℕ0 ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚)
25		r19.26 3046	. . . . . . . 8 ⊢ (∀𝑛 ∈ ℕ0 ((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛) ↔ (∀𝑛 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ∀𝑛 ∈ ℕ0 ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛))
26		simpllr 795	. . . . . . . . . 10 ⊢ ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚) ∧ (𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → 𝑚 ∈ ℕ0)
27		fveq2 6103	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = 𝑡 → ((𝑉 VDeg 𝐸)‘𝑢) = ((𝑉 VDeg 𝐸)‘𝑡))
28	27	eqeq1d 2612	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑡 → (((𝑉 VDeg 𝐸)‘𝑢) = 𝑚 ↔ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚))
29	28	rspcev 3282	. . . . . . . . . . . . . . 15 ⊢ ((𝑡 ∈ 𝑉 ∧ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚) → ∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑚)
30	29	adantlr 747	. . . . . . . . . . . . . 14 ⊢ (((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚) → ∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑚)
31	30	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚) ∧ (𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → ∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑚)
32		ornld 938	. . . . . . . . . . . . 13 ⊢ (∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑚 → (((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑚 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑚) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
33	31, 32	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚) ∧ (𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → (((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑚 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑚) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
34	33	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚) ∧ (𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) ∧ 𝑛 = 𝑚) → (((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑚 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑚) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
35		eqeq2 2621	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 ↔ ((𝑉 VDeg 𝐸)‘𝑢) = 𝑚))
36	35	rexbidv 3034	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 ↔ ∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑚))
37		breq2 4587	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ↔ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑚))
38	37	orbi1d 735	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → ((⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸) ↔ (⟨𝑉, 𝐸⟩ RegUSGrph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
39	36, 38	imbi12d 333	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → ((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ↔ (∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑚 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))))
40	37	notbid 307	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑚 → (¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ↔ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑚))
41	39, 40	anbi12d 743	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛) ↔ ((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑚 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑚)))
42	41	imbi1d 330	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸) ↔ (((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑚 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑚) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
43	42	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚) ∧ (𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) ∧ 𝑛 = 𝑚) → ((((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸) ↔ (((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑚 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑚 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑚) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
44	34, 43	mpbird 246	. . . . . . . . . 10 ⊢ (((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚) ∧ (𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) ∧ 𝑛 = 𝑚) → (((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
45	26, 44	rspcimdv 3283	. . . . . . . . 9 ⊢ ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚) ∧ (𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → (∀𝑛 ∈ ℕ0 ((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
46	45	com12 32	. . . . . . . 8 ⊢ (∀𝑛 ∈ ℕ0 ((∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛) → ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚) ∧ (𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
47	25, 46	sylbir 224	. . . . . . 7 ⊢ ((∀𝑛 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) ∧ ∀𝑛 ∈ ℕ0 ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛) → ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚) ∧ (𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))
48	47	expcom 450	. . . . . 6 ⊢ (∀𝑛 ∈ ℕ0 ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 → (∀𝑛 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) → ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚) ∧ (𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
49	48	com13 86	. . . . 5 ⊢ ((((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) ∧ ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚) ∧ (𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → (∀𝑛 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) → (∀𝑛 ∈ ℕ0 ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
50	49	exp31 628	. . . 4 ⊢ ((𝑡 ∈ 𝑉 ∧ 𝑚 ∈ ℕ0) → (((𝑉 VDeg 𝐸)‘𝑡) = 𝑚 → ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → (∀𝑛 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) → (∀𝑛 ∈ ℕ0 ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))))
51	50	rexlimivv 3018	. . 3 ⊢ (∃𝑡 ∈ 𝑉 ∃𝑚 ∈ ℕ0 ((𝑉 VDeg 𝐸)‘𝑡) = 𝑚 → ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → (∀𝑛 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) → (∀𝑛 ∈ ℕ0 ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))))
52	24, 51	mpcom 37	. 2 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → (∀𝑛 ∈ ℕ0 (∃𝑢 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑢) = 𝑛 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)) → (∀𝑛 ∈ ℕ0 ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑛 → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))
53	2, 3, 52	mp2d 47	1 ⊢ ((𝑉 FriendGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537  ∅c0 3874  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℝcr 9814  0cc0 9815   < clt 9953  3c3 10948  ℕ₀cn0 11169  #chash 12979   USGrph cusg 25859   VDeg cvdg 26420   RegUSGrph crusgra 26450   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-inf2 8421  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  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  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-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  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-se 4998  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-isom 5813  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-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  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-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-xadd 11823  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-reps 13161  df-csh 13386  df-s2 13444  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-dvds 14822  df-gcd 15055  df-prm 15224  df-phi 15309  df-usgra 25862  df-nbgra 25949  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045  df-wwlk 26207  df-wwlkn 26208  df-clwwlk 26279  df-clwwlkn 26280  df-2wlkonot 26385  df-2spthonot 26387  df-2spthsot 26388  df-vdgr 26421  df-rgra 26451  df-rusgra 26452  df-frgra 26516

This theorem is referenced by:  friendship  26649