Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frgrregorufr0 Structured version   Visualization version   GIF version

Theorem frgrregorufr0 41489
 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.)
Hypotheses
Ref Expression
frgrregorufr0.v 𝑉 = (Vtx‘𝐺)
frgrregorufr0.e 𝐸 = (Edg‘𝐺)
frgrregorufr0.d 𝐷 = (VtxDeg‘𝐺)
Assertion
Ref Expression
frgrregorufr0 (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))
Distinct variable groups:   𝑣,𝐷,𝑤   𝑣,𝐸   𝑤,𝐺   𝑣,𝐾,𝑤   𝑣,𝑉,𝑤
Allowed substitution hints:   𝐸(𝑤)   𝐺(𝑣)

Proof of Theorem frgrregorufr0
Dummy variables 𝑟 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgrregorufr0.v . . . 4 𝑉 = (Vtx‘𝐺)
2 frgrregorufr0.d . . . 4 𝐷 = (VtxDeg‘𝐺)
3 fveq2 6103 . . . . . 6 (𝑣 = 𝑡 → (𝐷𝑣) = (𝐷𝑡))
43eqeq1d 2612 . . . . 5 (𝑣 = 𝑡 → ((𝐷𝑣) = 𝐾 ↔ (𝐷𝑡) = 𝐾))
54cbvrabv 3172 . . . 4 {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = {𝑡𝑉 ∣ (𝐷𝑡) = 𝐾}
6 eqid 2610 . . . 4 (𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = (𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})
7 frgrregorufr0.e . . . 4 𝐸 = (Edg‘𝐺)
81, 2, 5, 6, 7frgrwopreg 41486 . . 3 (𝐺 ∈ FriendGraph → (((#‘{𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = 1 ∨ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = ∅) ∨ ((#‘(𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})) = 1 ∨ (𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = ∅)))
9 fveq2 6103 . . . . . . . . . 10 (𝑣 = 𝑟 → (𝐷𝑣) = (𝐷𝑟))
109eqeq1d 2612 . . . . . . . . 9 (𝑣 = 𝑟 → ((𝐷𝑣) = 𝐾 ↔ (𝐷𝑟) = 𝐾))
1110cbvrabv 3172 . . . . . . . 8 {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = {𝑟𝑉 ∣ (𝐷𝑟) = 𝐾}
12 fveq2 6103 . . . . . . . . . . 11 (𝑠 = 𝑣 → (𝐷𝑠) = (𝐷𝑣))
1312eqeq1d 2612 . . . . . . . . . 10 (𝑠 = 𝑣 → ((𝐷𝑠) = 𝐾 ↔ (𝐷𝑣) = 𝐾))
1413cbvrabv 3172 . . . . . . . . 9 {𝑠𝑉 ∣ (𝐷𝑠) = 𝐾} = {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}
1514difeq2i 3687 . . . . . . . 8 (𝑉 ∖ {𝑠𝑉 ∣ (𝐷𝑠) = 𝐾}) = (𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})
161, 2, 11, 15, 7frgrwopreg1 41487 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (#‘{𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = 1) → ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)
17163mix3d 1231 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (#‘{𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = 1) → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸))
1817expcom 450 . . . . 5 ((#‘{𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = 1 → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
19 rabeq0 3911 . . . . . 6 ({𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = ∅ ↔ ∀𝑣𝑉 ¬ (𝐷𝑣) = 𝐾)
20 neqne 2790 . . . . . . . . 9 (¬ (𝐷𝑣) = 𝐾 → (𝐷𝑣) ≠ 𝐾)
2120ralimi 2936 . . . . . . . 8 (∀𝑣𝑉 ¬ (𝐷𝑣) = 𝐾 → ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾)
22213mix2d 1230 . . . . . . 7 (∀𝑣𝑉 ¬ (𝐷𝑣) = 𝐾 → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸))
2322a1d 25 . . . . . 6 (∀𝑣𝑉 ¬ (𝐷𝑣) = 𝐾 → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
2419, 23sylbi 206 . . . . 5 ({𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = ∅ → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
2518, 24jaoi 393 . . . 4 (((#‘{𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = 1 ∨ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = ∅) → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
26 fveq2 6103 . . . . . . . . . 10 (𝑟 = 𝑠 → (𝐷𝑟) = (𝐷𝑠))
2726eqeq1d 2612 . . . . . . . . 9 (𝑟 = 𝑠 → ((𝐷𝑟) = 𝐾 ↔ (𝐷𝑠) = 𝐾))
2827cbvrabv 3172 . . . . . . . 8 {𝑟𝑉 ∣ (𝐷𝑟) = 𝐾} = {𝑠𝑉 ∣ (𝐷𝑠) = 𝐾}
2911difeq2i 3687 . . . . . . . 8 (𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = (𝑉 ∖ {𝑟𝑉 ∣ (𝐷𝑟) = 𝐾})
301, 2, 28, 29, 7frgrwopreg2 41488 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (#‘(𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})) = 1) → ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)
31303mix3d 1231 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (#‘(𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})) = 1) → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸))
3231expcom 450 . . . . 5 ((#‘(𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})) = 1 → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
33 difrab0eq 3990 . . . . . 6 ((𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = ∅ ↔ 𝑉 = {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})
34 rabid2 3096 . . . . . . 7 (𝑉 = {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} ↔ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)
35 3mix1 1223 . . . . . . . 8 (∀𝑣𝑉 (𝐷𝑣) = 𝐾 → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸))
3635a1d 25 . . . . . . 7 (∀𝑣𝑉 (𝐷𝑣) = 𝐾 → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
3734, 36sylbi 206 . . . . . 6 (𝑉 = {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
3833, 37sylbi 206 . . . . 5 ((𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = ∅ → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
3932, 38jaoi 393 . . . 4 (((#‘(𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})) = 1 ∨ (𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = ∅) → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
4025, 39jaoi 393 . . 3 ((((#‘{𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = 1 ∨ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = ∅) ∨ ((#‘(𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})) = 1 ∨ (𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = ∅)) → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
418, 40mpcom 37 . 2 (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸))
42 biidd 251 . . 3 (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ↔ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))
43 biidd 251 . . 3 (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ↔ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾))
44 sneq 4135 . . . . . . 7 (𝑣 = 𝑡 → {𝑣} = {𝑡})
4544difeq2d 3690 . . . . . 6 (𝑣 = 𝑡 → (𝑉 ∖ {𝑣}) = (𝑉 ∖ {𝑡}))
46 preq1 4212 . . . . . . 7 (𝑣 = 𝑡 → {𝑣, 𝑤} = {𝑡, 𝑤})
4746eleq1d 2672 . . . . . 6 (𝑣 = 𝑡 → ({𝑣, 𝑤} ∈ 𝐸 ↔ {𝑡, 𝑤} ∈ 𝐸))
4845, 47raleqbidv 3129 . . . . 5 (𝑣 = 𝑡 → (∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸 ↔ ∀𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸))
4948cbvrexv 3148 . . . 4 (∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)
5049a1i 11 . . 3 (𝐺 ∈ FriendGraph → (∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸 ↔ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸))
5142, 43, 503orbi123d 1390 . 2 (𝐺 ∈ FriendGraph → ((∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸) ↔ (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
5241, 51mpbird 246 1 (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537  ∅c0 3874  {csn 4125  {cpr 4127  ‘cfv 5804  1c1 9816  #chash 12979  Vtxcvtx 25673  Edgcedga 25792  VtxDegcvtxdg 40681   FriendGraph cfrgr 41428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-uhgr 25724  df-ushgr 25725  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-nbgr 40554  df-vtxdg 40682  df-frgr 41429 This theorem is referenced by:  frgrregorufr  41490
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