Theorem frgrregorufr0 30790

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 6891	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑦) = 𝐾))
4	3	cbvrabv 3424	. . 3 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = {𝑦 ∈ 𝑉 ∣ (𝐷‘𝑦) = 𝐾}
5		eqid 2762	. . 3 ⊢ (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}) = (𝑉 ∖ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾})
6	1, 2, 4, 5	frgrwopreg 30789	. 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 30784	. . . . . 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 6891	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → ((𝐷‘𝑥) = 𝐾 ↔ (𝐷‘𝑣) = 𝐾))
12	11	cbvrabv 3424	. . . . . . 7 ⊢ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} = {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾}
13	12	eqeq1i 2767	. . . . . 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 2965	. . . . . . . 8 ⊢ (¬ (𝐷‘𝑣) = 𝐾 → (𝐷‘𝑣) ≠ 𝐾)
17	16	ralimi 3101	. . . . . . 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 30785	. . . . . 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 2775	. . . . . . 7 ⊢ (𝑉 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} ↔ 𝑉 = {𝑣 ∈ 𝑉 ∣ (𝐷‘𝑣) = 𝐾})
27		rabid2 3447	. . . . . . 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 2145   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3414   ∖ cdif 3899  ∅c0 4282  {csn 4587  {cpr 4589  ‘cfv 6537  1c1 11128  ♯chash 14396  Vtxcvtx 29439  Edgcedg 29490  VtxDegcvtxdg 29911   FriendGraph cfrgr 30724

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-oadd 8462  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9909  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-n0 12532  df-xnn0 12605  df-z 12619  df-uz 12891  df-xadd 13166  df-fz 13564  df-hash 14397  df-edg 29491  df-uhgr 29501  df-ushgr 29502  df-upgr 29525  df-umgr 29526  df-uspgr 29596  df-usgr 29597  df-nbgr 29779  df-vtxdg 29912  df-frgr 30725

This theorem is used by:  frgrregorufr  30791