Theorem frgrregorufr0 30955

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 6917	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑦) = 𝐾))
4	3	cbvrabv 3426	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = {𝑦 ∈ 𝑉 ∣ (𝐷‘𝑦) = 𝐾}
5		eqid 2763	. . 3 ⊢ (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})
6	1, 2, 4, 5	frgrwopreg 30954	. 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 30949	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ (♯‘{𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)
9	8	3mix3d 1357	. . . . 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 419	. . . 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 6917	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑣) = 𝐾))
12	11	cbvrabv 3426	. . . . . . 7 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾}
13	12	eqeq1i 2768	. . . . . 6 ⊢ ({𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = ∅ ↔ {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾} = ∅)
14		rabeq0 4341	. . . . . 6 ⊢ ({𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾} = ∅ ↔ ∀𝑣 ∈ 𝑉 ¬ (𝐷‘𝑣) = 𝐾)
15	13, 14	bitri 278	. . . . 5 ⊢ ({𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = ∅ ↔ ∀𝑣 ∈ 𝑉 ¬ (𝐷‘𝑣) = 𝐾)
16		neqne 2967	. . . . . . . 8 ⊢ (¬ (𝐷‘𝑣) = 𝐾 → (𝐷‘𝑣) ≠ 𝐾)
17	16	ralimi 3103	. . . . . . 7 ⊢ (∀𝑣 ∈ 𝑉 ¬ (𝐷‘𝑣) = 𝐾 → ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾)
18	17	3mix2d 1356	. . . . . 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 26	. . . . 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 220	. . . 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 871	. . 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 30950	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ (♯‘(𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)
23	22	3mix3d 1357	. . . . 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 419	. . . 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 4426	. . . . 5 ⊢ ((𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = ∅ ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})
26	12	eqeq2i 2776	. . . . . . 7 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ↔ 𝑉 = {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾})
27		rabid2 3448	. . . . . . 7 ⊢ (𝑉 = {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾} ↔ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)
28	26, 27	bitri 278	. . . . . 6 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ↔ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)
29		3mix1 1349	. . . . . . 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 26	. . . . . 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 220	. . . . 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 220	. . . 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 871	. . 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 871	. 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 39	1 ⊢ (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416   ∖ cdif 4048  ∅c0 4281  {csn 4588  {cpr 4590  ‘cfv 6558  1c1 11228  ♯chash 14501  Vtxcvtx 29604  Edgcedg 29655  VtxDegcvtxdg 30076   FriendGraph cfrgr 30889

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8745  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-dju 10009  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-n0 12634  df-xnn0 12707  df-z 12721  df-uz 12993  df-xadd 13269  df-fz 13667  df-hash 14502  df-edg 29656  df-uhgr 29666  df-ushgr 29667  df-upgr 29690  df-umgr 29691  df-uspgr 29761  df-usgr 29762  df-nbgr 29944  df-vtxdg 30077  df-frgr 30890

This theorem is used by:  frgrregorufr  30956