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Theorem friendshipgt3 27144
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)) → ∃𝑣𝑉𝑤 ∈ (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.
StepHypRef Expression
1 frgrreggt1.v . . . 4 𝑉 = (Vtx‘𝐺)
2 eqid 2621 . . . 4 (Edg‘𝐺) = (Edg‘𝐺)
31, 2frgrregorufrg 27097 . . 3 (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ0 (∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
433ad2ant1 1080 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∀𝑘 ∈ ℕ0 (∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
51frgrogt3nreg 27143 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘)
6 frgrusgr 27024 . . . . . . 7 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
76anim1i 591 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
81isfusgr 26132 . . . . . 6 (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin))
97, 8sylibr 224 . . . . 5 ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph )
1093adant3 1079 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 𝐺 ∈ FinUSGraph )
11 0red 10001 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 0 ∈ ℝ)
12 3re 11054 . . . . . . . . 9 3 ∈ ℝ
1312a1i 11 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 3 ∈ ℝ)
14 hashcl 13103 . . . . . . . . . 10 (𝑉 ∈ Fin → (#‘𝑉) ∈ ℕ0)
1514nn0red 11312 . . . . . . . . 9 (𝑉 ∈ Fin → (#‘𝑉) ∈ ℝ)
1615adantr 481 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → (#‘𝑉) ∈ ℝ)
17 3pos 11074 . . . . . . . . 9 0 < 3
1817a1i 11 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 0 < 3)
19 simpr 477 . . . . . . . 8 ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 3 < (#‘𝑉))
2011, 13, 16, 18, 19lttrd 10158 . . . . . . 7 ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 0 < (#‘𝑉))
2120gt0ne0d 10552 . . . . . 6 ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → (#‘𝑉) ≠ 0)
22 hasheq0 13110 . . . . . . . 8 (𝑉 ∈ Fin → ((#‘𝑉) = 0 ↔ 𝑉 = ∅))
2322adantr 481 . . . . . . 7 ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ((#‘𝑉) = 0 ↔ 𝑉 = ∅))
2423necon3bid 2834 . . . . . 6 ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ((#‘𝑉) ≠ 0 ↔ 𝑉 ≠ ∅))
2521, 24mpbid 222 . . . . 5 ((𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 𝑉 ≠ ∅)
26253adant1 1077 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → 𝑉 ≠ ∅)
271fusgrn0degnn0 26315 . . . 4 ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → ∃𝑡𝑉𝑚 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑡) = 𝑚)
2810, 26, 27syl2anc 692 . . 3 ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∃𝑡𝑉𝑚 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑡) = 𝑚)
29 r19.26 3059 . . . . . . . 8 (∀𝑘 ∈ ℕ0 ((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) ↔ (∀𝑘 ∈ ℕ0 (∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘))
30 simpllr 798 . . . . . . . . . 10 ((((𝑡𝑉𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → 𝑚 ∈ ℕ0)
31 fveq2 6158 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑡 → ((VtxDeg‘𝐺)‘𝑢) = ((VtxDeg‘𝐺)‘𝑡))
3231eqeq1d 2623 . . . . . . . . . . . . . . 15 (𝑢 = 𝑡 → (((VtxDeg‘𝐺)‘𝑢) = 𝑚 ↔ ((VtxDeg‘𝐺)‘𝑡) = 𝑚))
3332rspcev 3299 . . . . . . . . . . . . . 14 ((𝑡𝑉 ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) → ∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚)
3433ad4ant13 1289 . . . . . . . . . . . . 13 ((((𝑡𝑉𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → ∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚)
35 ornld 939 . . . . . . . . . . . . 13 (∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑚) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
3634, 35syl 17 . . . . . . . . . . . 12 ((((𝑡𝑉𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → (((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑚) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
3736adantr 481 . . . . . . . . . . 11 (((((𝑡𝑉𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) ∧ 𝑘 = 𝑚) → (((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑚) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
38 eqeq2 2632 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (((VtxDeg‘𝐺)‘𝑢) = 𝑘 ↔ ((VtxDeg‘𝐺)‘𝑢) = 𝑚))
3938rexbidv 3047 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 ↔ ∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚))
40 breq2 4627 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (𝐺 RegUSGraph 𝑘𝐺 RegUSGraph 𝑚))
4140orbi1d 738 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) ↔ (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
4239, 41imbi12d 334 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → ((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ↔ (∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))))
4340notbid 308 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → (¬ 𝐺 RegUSGraph 𝑘 ↔ ¬ 𝐺 RegUSGraph 𝑚))
4442, 43anbi12d 746 . . . . . . . . . . . . 13 (𝑘 = 𝑚 → (((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) ↔ ((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑚)))
4544imbi1d 331 . . . . . . . . . . . 12 (𝑘 = 𝑚 → ((((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) ↔ (((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑚) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
4645adantl 482 . . . . . . . . . . 11 (((((𝑡𝑉𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) ∧ 𝑘 = 𝑚) → ((((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)) ↔ (((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑚 → (𝐺 RegUSGraph 𝑚 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑚) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
4737, 46mpbird 247 . . . . . . . . . 10 (((((𝑡𝑉𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) ∧ 𝑘 = 𝑚) → (((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
4830, 47rspcimdv 3300 . . . . . . . . 9 ((((𝑡𝑉𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → (∀𝑘 ∈ ℕ0 ((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
4948com12 32 . . . . . . . 8 (∀𝑘 ∈ ℕ0 ((∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ¬ 𝐺 RegUSGraph 𝑘) → ((((𝑡𝑉𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
5029, 49sylbir 225 . . . . . . 7 ((∀𝑘 ∈ ℕ0 (∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) ∧ ∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘) → ((((𝑡𝑉𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))
5150expcom 451 . . . . . 6 (∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘 → (∀𝑘 ∈ ℕ0 (∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) → ((((𝑡𝑉𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
5251com13 88 . . . . 5 ((((𝑡𝑉𝑚 ∈ ℕ0) ∧ ((VtxDeg‘𝐺)‘𝑡) = 𝑚) ∧ (𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉))) → (∀𝑘 ∈ ℕ0 (∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) → (∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘 → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
5352exp31 629 . . . 4 ((𝑡𝑉𝑚 ∈ ℕ0) → (((VtxDeg‘𝐺)‘𝑡) = 𝑚 → ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → (∀𝑘 ∈ ℕ0 (∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) → (∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘 → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))))
5453rexlimivv 3031 . . 3 (∃𝑡𝑉𝑚 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑡) = 𝑚 → ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → (∀𝑘 ∈ ℕ0 (∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) → (∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘 → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺)))))
5528, 54mpcom 38 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → (∀𝑘 ∈ ℕ0 (∃𝑢𝑉 ((VtxDeg‘𝐺)‘𝑢) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))) → (∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘 → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))))
564, 5, 55mp2d 49 1 ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  cdif 3557  c0 3897  {csn 4155  {cpr 4157   class class class wbr 4623  cfv 5857  Fincfn 7915  cr 9895  0cc0 9896   < clt 10034  3c3 11031  0cn0 11252  #chash 13073  Vtxcvtx 25808  Edgcedg 25873   USGraph cusgr 25971   FinUSGraph cfusgr 26130  VtxDegcvtxdg 26282   RegUSGraph crusgr 26356   FriendGraph cfrgr 27020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-ac2 9245  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-disj 4594  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-ec 7704  df-qs 7708  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-ac 8899  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-xnn0 11324  df-z 11338  df-uz 11648  df-rp 11793  df-xadd 11907  df-ico 12139  df-fz 12285  df-fzo 12423  df-fl 12549  df-mod 12625  df-seq 12758  df-exp 12817  df-hash 13074  df-word 13254  df-lsw 13255  df-concat 13256  df-s1 13257  df-substr 13258  df-reps 13261  df-csh 13488  df-s2 13546  df-s3 13547  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-sum 14367  df-dvds 14927  df-gcd 15160  df-prm 15329  df-phi 15414  df-vtx 25810  df-iedg 25811  df-edg 25874  df-uhgr 25883  df-ushgr 25884  df-upgr 25907  df-umgr 25908  df-uspgr 25972  df-usgr 25973  df-fusgr 26131  df-nbgr 26149  df-vtxdg 26283  df-rgr 26357  df-rusgr 26358  df-wlks 26399  df-wlkson 26400  df-trls 26492  df-trlson 26493  df-pths 26515  df-spths 26516  df-pthson 26517  df-spthson 26518  df-wwlks 26625  df-wwlksn 26626  df-wwlksnon 26627  df-wspthsn 26628  df-wspthsnon 26629  df-clwwlks 26778  df-clwwlksn 26779  df-conngr 26947  df-frgr 27021
This theorem is referenced by:  friendship  27145
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