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Theorem frgrregorufr0 27304
Description: In a friendship graph there are either no vertices having degree 𝐾, or all vertices have degree 𝐾 for any (nonnegative integer) 𝐾, unless there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "... all vertices have degree k, unless there is a universal friend." (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Revised by AV, 11-May-2021.) (Proof shortened by AV, 3-Jan-2022.)
Hypotheses
Ref Expression
frgrregorufr0.v 𝑉 = (Vtx‘𝐺)
frgrregorufr0.e 𝐸 = (Edg‘𝐺)
frgrregorufr0.d 𝐷 = (VtxDeg‘𝐺)
Assertion
Ref Expression
frgrregorufr0 (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))
Distinct variable groups:   𝑣,𝐷,𝑤   𝑣,𝐸   𝑣,𝐺,𝑤   𝑣,𝐾,𝑤   𝑣,𝑉,𝑤
Allowed substitution hint:   𝐸(𝑤)

Proof of Theorem frgrregorufr0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrregorufr0.v . . 3 𝑉 = (Vtx‘𝐺)
2 frgrregorufr0.d . . 3 𝐷 = (VtxDeg‘𝐺)
3 fveq2 6229 . . . . 5 (𝑥 = 𝑦 → (𝐷𝑥) = (𝐷𝑦))
43eqeq1d 2653 . . . 4 (𝑥 = 𝑦 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑦) = 𝐾))
54cbvrabv 3230 . . 3 {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} = {𝑦𝑉 ∣ (𝐷𝑦) = 𝐾}
6 eqid 2651 . . 3 (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾})
71, 2, 5, 6frgrwopreg 27303 . 2 (𝐺 ∈ FriendGraph → (((#‘{𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = 1 ∨ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} = ∅) ∨ ((#‘(𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾})) = 1 ∨ (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = ∅)))
8 frgrregorufr0.e . . . . . . 7 𝐸 = (Edg‘𝐺)
91, 2, 5, 6, 8frgrwopreg1 27298 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (#‘{𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = 1) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)
1093mix3d 1258 . . . . 5 ((𝐺 ∈ FriendGraph ∧ (#‘{𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = 1) → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))
1110expcom 450 . . . 4 ((#‘{𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = 1 → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
12 fveq2 6229 . . . . . . . . 9 (𝑥 = 𝑣 → (𝐷𝑥) = (𝐷𝑣))
1312eqeq1d 2653 . . . . . . . 8 (𝑥 = 𝑣 → ((𝐷𝑥) = 𝐾 ↔ (𝐷𝑣) = 𝐾))
1413cbvrabv 3230 . . . . . . 7 {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} = {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}
1514eqeq1i 2656 . . . . . 6 ({𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} = ∅ ↔ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = ∅)
16 rabeq0 3990 . . . . . 6 ({𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = ∅ ↔ ∀𝑣𝑉 ¬ (𝐷𝑣) = 𝐾)
1715, 16bitri 264 . . . . 5 ({𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} = ∅ ↔ ∀𝑣𝑉 ¬ (𝐷𝑣) = 𝐾)
18 neqne 2831 . . . . . . . 8 (¬ (𝐷𝑣) = 𝐾 → (𝐷𝑣) ≠ 𝐾)
1918ralimi 2981 . . . . . . 7 (∀𝑣𝑉 ¬ (𝐷𝑣) = 𝐾 → ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾)
20193mix2d 1257 . . . . . 6 (∀𝑣𝑉 ¬ (𝐷𝑣) = 𝐾 → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))
2120a1d 25 . . . . 5 (∀𝑣𝑉 ¬ (𝐷𝑣) = 𝐾 → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
2217, 21sylbi 207 . . . 4 ({𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} = ∅ → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
2311, 22jaoi 393 . . 3 (((#‘{𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = 1 ∨ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} = ∅) → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
241, 2, 5, 6, 8frgrwopreg2 27299 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (#‘(𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾})) = 1) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)
25243mix3d 1258 . . . . 5 ((𝐺 ∈ FriendGraph ∧ (#‘(𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾})) = 1) → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))
2625expcom 450 . . . 4 ((#‘(𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾})) = 1 → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
27 difrab0eq 4071 . . . . 5 ((𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = ∅ ↔ 𝑉 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾})
2814eqeq2i 2663 . . . . . . 7 (𝑉 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} ↔ 𝑉 = {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})
29 rabid2 3148 . . . . . . 7 (𝑉 = {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} ↔ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)
3028, 29bitri 264 . . . . . 6 (𝑉 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} ↔ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)
31 3mix1 1250 . . . . . . 7 (∀𝑣𝑉 (𝐷𝑣) = 𝐾 → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))
3231a1d 25 . . . . . 6 (∀𝑣𝑉 (𝐷𝑣) = 𝐾 → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
3330, 32sylbi 207 . . . . 5 (𝑉 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
3427, 33sylbi 207 . . . 4 ((𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = ∅ → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
3526, 34jaoi 393 . . 3 (((#‘(𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾})) = 1 ∨ (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = ∅) → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
3623, 35jaoi 393 . 2 ((((#‘{𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = 1 ∨ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} = ∅) ∨ ((#‘(𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾})) = 1 ∨ (𝑉 ∖ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}) = ∅)) → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
377, 36mpcom 38 1 (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  w3o 1053   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  cdif 3604  c0 3948  {csn 4210  {cpr 4212  cfv 5926  1c1 9975  #chash 13157  Vtxcvtx 25919  Edgcedg 25984  VtxDegcvtxdg 26417   FriendGraph cfrgr 27236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-xadd 11985  df-fz 12365  df-hash 13158  df-edg 25985  df-uhgr 25998  df-ushgr 25999  df-upgr 26022  df-umgr 26023  df-uspgr 26090  df-usgr 26091  df-nbgr 26270  df-vtxdg 26418  df-frgr 27237
This theorem is referenced by:  frgrregorufr  27305
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