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Theorem frgrregorufr0 27054
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 6150 . . . . . 6 (𝑣 = 𝑡 → (𝐷𝑣) = (𝐷𝑡))
43eqeq1d 2623 . . . . 5 (𝑣 = 𝑡 → ((𝐷𝑣) = 𝐾 ↔ (𝐷𝑡) = 𝐾))
54cbvrabv 3185 . . . 4 {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = {𝑡𝑉 ∣ (𝐷𝑡) = 𝐾}
6 eqid 2621 . . . 4 (𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = (𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})
7 frgrregorufr0.e . . . 4 𝐸 = (Edg‘𝐺)
81, 2, 5, 6, 7frgrwopreg 27051 . . 3 (𝐺 ∈ FriendGraph → (((#‘{𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = 1 ∨ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = ∅) ∨ ((#‘(𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})) = 1 ∨ (𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = ∅)))
9 fveq2 6150 . . . . . . . . . 10 (𝑣 = 𝑟 → (𝐷𝑣) = (𝐷𝑟))
109eqeq1d 2623 . . . . . . . . 9 (𝑣 = 𝑟 → ((𝐷𝑣) = 𝐾 ↔ (𝐷𝑟) = 𝐾))
1110cbvrabv 3185 . . . . . . . 8 {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = {𝑟𝑉 ∣ (𝐷𝑟) = 𝐾}
12 fveq2 6150 . . . . . . . . . . 11 (𝑠 = 𝑣 → (𝐷𝑠) = (𝐷𝑣))
1312eqeq1d 2623 . . . . . . . . . 10 (𝑠 = 𝑣 → ((𝐷𝑠) = 𝐾 ↔ (𝐷𝑣) = 𝐾))
1413cbvrabv 3185 . . . . . . . . 9 {𝑠𝑉 ∣ (𝐷𝑠) = 𝐾} = {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}
1514difeq2i 3705 . . . . . . . 8 (𝑉 ∖ {𝑠𝑉 ∣ (𝐷𝑠) = 𝐾}) = (𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})
161, 2, 11, 15, 7frgrwopreg1 27052 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (#‘{𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = 1) → ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)
17163mix3d 1236 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (#‘{𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = 1) → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸))
1817expcom 451 . . . . 5 ((#‘{𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = 1 → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
19 rabeq0 3933 . . . . . 6 ({𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = ∅ ↔ ∀𝑣𝑉 ¬ (𝐷𝑣) = 𝐾)
20 neqne 2798 . . . . . . . . 9 (¬ (𝐷𝑣) = 𝐾 → (𝐷𝑣) ≠ 𝐾)
2120ralimi 2947 . . . . . . . 8 (∀𝑣𝑉 ¬ (𝐷𝑣) = 𝐾 → ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾)
22213mix2d 1235 . . . . . . 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 207 . . . . 5 ({𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = ∅ → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
2518, 24jaoi 394 . . . 4 (((#‘{𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = 1 ∨ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} = ∅) → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
26 fveq2 6150 . . . . . . . . . 10 (𝑟 = 𝑠 → (𝐷𝑟) = (𝐷𝑠))
2726eqeq1d 2623 . . . . . . . . 9 (𝑟 = 𝑠 → ((𝐷𝑟) = 𝐾 ↔ (𝐷𝑠) = 𝐾))
2827cbvrabv 3185 . . . . . . . 8 {𝑟𝑉 ∣ (𝐷𝑟) = 𝐾} = {𝑠𝑉 ∣ (𝐷𝑠) = 𝐾}
2911difeq2i 3705 . . . . . . . 8 (𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = (𝑉 ∖ {𝑟𝑉 ∣ (𝐷𝑟) = 𝐾})
301, 2, 28, 29, 7frgrwopreg2 27053 . . . . . . 7 ((𝐺 ∈ FriendGraph ∧ (#‘(𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})) = 1) → ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)
31303mix3d 1236 . . . . . 6 ((𝐺 ∈ FriendGraph ∧ (#‘(𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})) = 1) → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸))
3231expcom 451 . . . . 5 ((#‘(𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})) = 1 → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
33 difrab0eq 4012 . . . . . 6 ((𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = ∅ ↔ 𝑉 = {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾})
34 rabid2 3107 . . . . . . 7 (𝑉 = {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} ↔ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)
35 3mix1 1228 . . . . . . . 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 207 . . . . . 6 (𝑉 = {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾} → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
3833, 37sylbi 207 . . . . 5 ((𝑉 ∖ {𝑣𝑉 ∣ (𝐷𝑣) = 𝐾}) = ∅ → (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸)))
3932, 38jaoi 394 . . . 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 394 . . 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 38 . 2 (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑡𝑉𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸))
42 biidd 252 . . 3 (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ↔ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))
43 biidd 252 . . 3 (𝐺 ∈ FriendGraph → (∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ↔ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾))
44 sneq 4160 . . . . . . 7 (𝑣 = 𝑡 → {𝑣} = {𝑡})
4544difeq2d 3708 . . . . . 6 (𝑣 = 𝑡 → (𝑉 ∖ {𝑣}) = (𝑉 ∖ {𝑡}))
46 preq1 4240 . . . . . . 7 (𝑣 = 𝑡 → {𝑣, 𝑤} = {𝑡, 𝑤})
4746eleq1d 2683 . . . . . 6 (𝑣 = 𝑡 → ({𝑣, 𝑤} ∈ 𝐸 ↔ {𝑡, 𝑤} ∈ 𝐸))
4845, 47raleqbidv 3141 . . . . 5 (𝑣 = 𝑡 → (∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸 ↔ ∀𝑤 ∈ (𝑉 ∖ {𝑡}){𝑡, 𝑤} ∈ 𝐸))
4948cbvrexv 3160 . . . 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 1395 . 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 247 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 196  wo 383  wa 384  w3o 1035   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  cdif 3553  c0 3893  {csn 4150  {cpr 4152  cfv 5849  1c1 9884  #chash 13060  Vtxcvtx 25781  Edgcedg 25846  VtxDegcvtxdg 26255   FriendGraph cfrgr 26993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-card 8712  df-cda 8937  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-n0 11240  df-xnn0 11311  df-z 11325  df-uz 11635  df-xadd 11894  df-fz 12272  df-hash 13061  df-edg 25847  df-uhgr 25856  df-ushgr 25857  df-upgr 25880  df-umgr 25881  df-uspgr 25945  df-usgr 25946  df-nbgr 26122  df-vtxdg 26256  df-frgr 26994
This theorem is referenced by:  frgrregorufr  27055
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