Theorem frgrregorufr0 30413

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 6845	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑦) = 𝐾))
4	3	cbvrabv 3400	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = {𝑦 ∈ 𝑉 ∣ (𝐷‘𝑦) = 𝐾}
5		eqid 2737	. . 3 ⊢ (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})
6	1, 2, 4, 5	frgrwopreg 30412	. 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 30407	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ (♯‘{𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)
9	8	3mix3d 1340	. . . . 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 413	. . . 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 6845	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑣) = 𝐾))
12	11	cbvrabv 3400	. . . . . . 7 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾}
13	12	eqeq1i 2742	. . . . . 6 ⊢ ({𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = ∅ ↔ {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾} = ∅)
14		rabeq0 4329	. . . . . 6 ⊢ ({𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾} = ∅ ↔ ∀𝑣 ∈ 𝑉 ¬ (𝐷‘𝑣) = 𝐾)
15	13, 14	bitri 275	. . . . 5 ⊢ ({𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = ∅ ↔ ∀𝑣 ∈ 𝑉 ¬ (𝐷‘𝑣) = 𝐾)
16		neqne 2941	. . . . . . . 8 ⊢ (¬ (𝐷‘𝑣) = 𝐾 → (𝐷‘𝑣) ≠ 𝐾)
17	16	ralimi 3075	. . . . . . 7 ⊢ (∀𝑣 ∈ 𝑉 ¬ (𝐷‘𝑣) = 𝐾 → ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾)
18	17	3mix2d 1339	. . . . . 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 217	. . . 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 858	. . 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 30408	. . . . . 6 ⊢ ((𝐺 ∈ FriendGraph ∧ (♯‘(𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)
23	22	3mix3d 1340	. . . . 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 413	. . . 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 4411	. . . . 5 ⊢ ((𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = ∅ ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})
26	12	eqeq2i 2750	. . . . . . 7 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ↔ 𝑉 = {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾})
27		rabid2 3423	. . . . . . 7 ⊢ (𝑉 = {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾} ↔ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)
28	26, 27	bitri 275	. . . . . 6 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ↔ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)
29		3mix1 1332	. . . . . . 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 217	. . . . 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 217	. . . 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 858	. . 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 858	. 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 395   ∨ wo 848   ∨ w3o 1086   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390   ∖ cdif 3887  ∅c0 4274  {csn 4568  {cpr 4570  ‘cfv 6494  1c1 11034  ♯chash 14287  Vtxcvtx 29083  Edgcedg 29134  VtxDegcvtxdg 29553   FriendGraph cfrgr 30347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-oadd 8404  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-dju 9820  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-nn 12170  df-2 12239  df-n0 12433  df-xnn0 12506  df-z 12520  df-uz 12784  df-xadd 13059  df-fz 13457  df-hash 14288  df-edg 29135  df-uhgr 29145  df-ushgr 29146  df-upgr 29169  df-umgr 29170  df-uspgr 29237  df-usgr 29238  df-nbgr 29420  df-vtxdg 29554  df-frgr 30348

This theorem is referenced by:  frgrregorufr  30414