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Theorem frgrregorufrg 27707
 Description: If there is a vertex having degree 𝑘 for each nonnegative integer 𝑘 in a friendship graph, then there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "Suppose there is a vertex of degree k > 1. ... all vertices have degree k, unless there is a universal friend. ... It follows that G is k-regular, i.e., the degree of every vertex is k". Variant of frgrregorufr 27706 with generalization. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.)
Hypotheses
Ref Expression
frgrregorufrg.v 𝑉 = (Vtx‘𝐺)
frgrregorufrg.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frgrregorufrg (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ0 (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺RegUSGraph𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
Distinct variable groups:   𝐺,𝑎,𝑘,𝑣,𝑤   𝐸,𝑎,𝑣   𝑉,𝑎,𝑣,𝑤   𝑘,𝑎,𝑣,𝑤
Allowed substitution hints:   𝐸(𝑤,𝑘)   𝑉(𝑘)

Proof of Theorem frgrregorufrg
StepHypRef Expression
1 frgrregorufrg.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 frgrregorufrg.e . . . . 5 𝐸 = (Edg‘𝐺)
3 eqid 2825 . . . . 5 (VtxDeg‘𝐺) = (VtxDeg‘𝐺)
41, 2, 3frgrregorufr 27706 . . . 4 (𝐺 ∈ FriendGraph → (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
54adantr 474 . . 3 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
6 frgrusgr 27641 . . . . 5 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
7 nn0xnn0 11694 . . . . 5 (𝑘 ∈ ℕ0𝑘 ∈ ℕ0*)
81, 3usgreqdrusgr 26866 . . . . . 6 ((𝐺 ∈ USGraph ∧ 𝑘 ∈ ℕ0* ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘) → 𝐺RegUSGraph𝑘)
983expia 1156 . . . . 5 ((𝐺 ∈ USGraph ∧ 𝑘 ∈ ℕ0*) → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘𝐺RegUSGraph𝑘))
106, 7, 9syl2an 591 . . . 4 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘𝐺RegUSGraph𝑘))
1110orim1d 995 . . 3 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → ((∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) → (𝐺RegUSGraph𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
125, 11syld 47 . 2 ((𝐺 ∈ FriendGraph ∧ 𝑘 ∈ ℕ0) → (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺RegUSGraph𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
1312ralrimiva 3175 1 (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ0 (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺RegUSGraph𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∨ wo 880   = wceq 1658   ∈ wcel 2166  ∀wral 3117  ∃wrex 3118   ∖ cdif 3795  {csn 4397  {cpr 4399   class class class wbr 4873  ‘cfv 6123  ℕ0cn0 11618  ℕ0*cxnn0 11690  Vtxcvtx 26294  Edgcedg 26345  USGraphcusgr 26448  VtxDegcvtxdg 26763  RegUSGraphcrusgr 26854   FriendGraph cfrgr 27637 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-card 9078  df-cda 9305  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-n0 11619  df-xnn0 11691  df-z 11705  df-uz 11969  df-xadd 12233  df-fz 12620  df-hash 13411  df-edg 26346  df-uhgr 26356  df-ushgr 26357  df-upgr 26380  df-umgr 26381  df-uspgr 26449  df-usgr 26450  df-nbgr 26630  df-vtxdg 26764  df-rgr 26855  df-rusgr 26856  df-frgr 27638 This theorem is referenced by:  friendshipgt3  27813
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