Theorem frgrregorufr0 28103

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 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (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.

Step	Hyp	Ref	Expression
1		frgrregorufr0.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		frgrregorufr0.d	. . 3 ⊢ 𝐷 = (VtxDeg‘𝐺)
3		fveqeq2 6679	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑦) = 𝐾))
4	3	cbvrabv 3491	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = {𝑦 ∈ 𝑉 ∣ (𝐷‘𝑦) = 𝐾}
5		eqid 2821	. . 3 ⊢ (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})
6	1, 2, 4, 5	frgrwopreg 28102	. 2 ⊢ (𝐺 ∈ FriendGraph → (((♯‘{𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = 1 ∨ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = ∅) ∨ ((♯‘(𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})) = 1 ∨ (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = ∅)))
7		frgrregorufr0.e	. . . . . . 7 ⊢ 𝐸 = (Edg‘𝐺)
8	1, 2, 4, 5, 7	frgrwopreg1 28097	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ (♯‘{𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)
9	8	3mix3d 1334	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ (♯‘{𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = 1) → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))
10	9	expcom 416	. . . 4 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = 1 → (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
11		fveqeq2 6679	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑣) = 𝐾))
12	11	cbvrabv 3491	. . . . . . 7 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾}
13	12	eqeq1i 2826	. . . . . 6 ⊢ ({𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = ∅ ↔ {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾} = ∅)
14		rabeq0 4338	. . . . . 6 ⊢ ({𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾} = ∅ ↔ ∀𝑣 ∈ 𝑉 ¬ (𝐷‘𝑣) = 𝐾)
15	13, 14	bitri 277	. . . . 5 ⊢ ({𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = ∅ ↔ ∀𝑣 ∈ 𝑉 ¬ (𝐷‘𝑣) = 𝐾)
16		neqne 3024	. . . . . . . 8 ⊢ (¬ (𝐷‘𝑣) = 𝐾 → (𝐷‘𝑣) ≠ 𝐾)
17	16	ralimi 3160	. . . . . . 7 ⊢ (∀𝑣 ∈ 𝑉 ¬ (𝐷‘𝑣) = 𝐾 → ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾)
18	17	3mix2d 1333	. . . . . 6 ⊢ (∀𝑣 ∈ 𝑉 ¬ (𝐷‘𝑣) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))
19	18	a1d 25	. . . . 5 ⊢ (∀𝑣 ∈ 𝑉 ¬ (𝐷‘𝑣) = 𝐾 → (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
20	15, 19	sylbi 219	. . . 4 ⊢ ({𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = ∅ → (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
21	10, 20	jaoi 853	. . 3 ⊢ (((♯‘{𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = 1 ∨ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = ∅) → (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
22	1, 2, 4, 5, 7	frgrwopreg2 28098	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ (♯‘(𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)
23	22	3mix3d 1334	. . . . 5 ⊢ ((𝐺 ∈ FriendGraph ∧ (♯‘(𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})) = 1) → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))
24	23	expcom 416	. . . 4 ⊢ ((♯‘(𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})) = 1 → (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
25		difrab0eq 4419	. . . . 5 ⊢ ((𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = ∅ ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})
26	12	eqeq2i 2834	. . . . . . 7 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ↔ 𝑉 = {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾})
27		rabid2 3381	. . . . . . 7 ⊢ (𝑉 = {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾} ↔ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)
28	26, 27	bitri 277	. . . . . 6 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ↔ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)
29		3mix1 1326	. . . . . . 7 ⊢ (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))
30	29	a1d 25	. . . . . 6 ⊢ (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
31	28, 30	sylbi 219	. . . . 5 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} → (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
32	25, 31	sylbi 219	. . . 4 ⊢ ((𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = ∅ → (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
33	24, 32	jaoi 853	. . 3 ⊢ (((♯‘(𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})) = 1 ∨ (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = ∅) → (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
34	21, 33	jaoi 853	. 2 ⊢ ((((♯‘{𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = 1 ∨ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = ∅) ∨ ((♯‘(𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})) = 1 ∨ (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = ∅)) → (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))
35	6, 34	mpcom 38	1 ⊢ (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   ∨ w3o 1082   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142   ∖ cdif 3933  ∅c0 4291  {csn 4567  {cpr 4569  ‘cfv 6355  1c1 10538  ♯chash 13691  Vtxcvtx 26781  Edgcedg 26832  VtxDegcvtxdg 27247   FriendGraph cfrgr 28037

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 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  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 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-dju 9330  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-n0 11899  df-xnn0 11969  df-z 11983  df-uz 12245  df-xadd 12509  df-fz 12894  df-hash 13692  df-edg 26833  df-uhgr 26843  df-ushgr 26844  df-upgr 26867  df-umgr 26868  df-uspgr 26935  df-usgr 26936  df-nbgr 27115  df-vtxdg 27248  df-frgr 28038

This theorem is referenced by:  frgrregorufr  28104