Theorem frgrregorufr0 30394

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 6849	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑦) = 𝐾))
4	3	cbvrabv 3399	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = {𝑦 ∈ 𝑉 ∣ (𝐷‘𝑦) = 𝐾}
5		eqid 2736	. . 3 ⊢ (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})
6	1, 2, 4, 5	frgrwopreg 30393	. 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 30388	. . . . . 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 6849	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑣) = 𝐾))
12	11	cbvrabv 3399	. . . . . . 7 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾}
13	12	eqeq1i 2741	. . . . . 6 ⊢ ({𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = ∅ ↔ {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾} = ∅)
14		rabeq0 4328	. . . . . 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 2940	. . . . . . . 8 ⊢ (¬ (𝐷‘𝑣) = 𝐾 → (𝐷‘𝑣) ≠ 𝐾)
17	16	ralimi 3074	. . . . . . 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 30389	. . . . . 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 4410	. . . . 5 ⊢ ((𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = ∅ ↔ 𝑉 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})
26	12	eqeq2i 2749	. . . . . . 7 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ↔ 𝑉 = {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾})
27		rabid2 3422	. . . . . . 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 2932  ∀wral 3051  ∃wrex 3061  {crab 3389   ∖ cdif 3886  ∅c0 4273  {csn 4567  {cpr 4569  ‘cfv 6498  1c1 11039  ♯chash 14292  Vtxcvtx 29065  Edgcedg 29116  VtxDegcvtxdg 29534   FriendGraph cfrgr 30328

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-xadd 13064  df-fz 13462  df-hash 14293  df-edg 29117  df-uhgr 29127  df-ushgr 29128  df-upgr 29151  df-umgr 29152  df-uspgr 29219  df-usgr 29220  df-nbgr 29402  df-vtxdg 29535  df-frgr 30329

This theorem is referenced by:  frgrregorufr  30395