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Theorem rusgrasn 26472
 Description: If a k-regular undirected simple graph has only one vertex, then k must be 0. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
Assertion
Ref Expression
rusgrasn (((#‘𝑉) = 1 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → 𝐾 = 0)

Proof of Theorem rusgrasn
Dummy variables 𝑘 𝑛 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rusgraprop3 26470 . . 3 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑘𝑉 (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾))
2 usgrav 25867 . . . . 5 (𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3 hash1snb 13068 . . . . . . . 8 (𝑉 ∈ V → ((#‘𝑉) = 1 ↔ ∃𝑣 𝑉 = {𝑣}))
4 raleq 3115 . . . . . . . . . . . . 13 (𝑉 = {𝑣} → (∀𝑘𝑉 (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾 ↔ ∀𝑘 ∈ {𝑣} (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾))
5 vex 3176 . . . . . . . . . . . . . 14 𝑣 ∈ V
6 preq1 4212 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑣 → {𝑘, 𝑛} = {𝑣, 𝑛})
76eleq1d 2672 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑣 → ({𝑘, 𝑛} ∈ ran 𝐸 ↔ {𝑣, 𝑛} ∈ ran 𝐸))
87rabbidv 3164 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑣 → {𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸} = {𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸})
98fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑣 → (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}))
109eqeq1d 2612 . . . . . . . . . . . . . . 15 (𝑘 = 𝑣 → ((#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾 ↔ (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾))
1110ralsng 4165 . . . . . . . . . . . . . 14 (𝑣 ∈ V → (∀𝑘 ∈ {𝑣} (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾 ↔ (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾))
125, 11mp1i 13 . . . . . . . . . . . . 13 (𝑉 = {𝑣} → (∀𝑘 ∈ {𝑣} (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾 ↔ (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾))
13 rabeq 3166 . . . . . . . . . . . . . . 15 (𝑉 = {𝑣} → {𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸} = {𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸})
1413fveq2d 6107 . . . . . . . . . . . . . 14 (𝑉 = {𝑣} → (#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = (#‘{𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸}))
1514eqeq1d 2612 . . . . . . . . . . . . 13 (𝑉 = {𝑣} → ((#‘{𝑛𝑉 ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 ↔ (#‘{𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾))
164, 12, 153bitrd 293 . . . . . . . . . . . 12 (𝑉 = {𝑣} → (∀𝑘𝑉 (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾 ↔ (#‘{𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾))
17 hashrabsn01 13023 . . . . . . . . . . . . . 14 ((#‘{𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 → (𝐾 = 0 ∨ 𝐾 = 1))
18 ax-1 6 . . . . . . . . . . . . . . . . 17 (𝐾 = 0 → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0))
19182a1d 26 . . . . . . . . . . . . . . . 16 (𝐾 = 0 → (𝐾 ∈ ℕ0 → (𝑉 ∈ V → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0))))
20192a1d 26 . . . . . . . . . . . . . . 15 (𝐾 = 0 → ((#‘{𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 → (𝑉 = {𝑣} → (𝐾 ∈ ℕ0 → (𝑉 ∈ V → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0))))))
21 eqeq2 2621 . . . . . . . . . . . . . . . 16 (𝐾 = 1 → ((#‘{𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 ↔ (#‘{𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 1))
22 hashrabsn1 13024 . . . . . . . . . . . . . . . . 17 ((#‘{𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 1 → [𝑣 / 𝑛]{𝑣, 𝑛} ∈ ran 𝐸)
23 sbcel1g 3939 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ V → ([𝑣 / 𝑛]{𝑣, 𝑛} ∈ ran 𝐸𝑣 / 𝑛{𝑣, 𝑛} ∈ ran 𝐸))
245, 23ax-mp 5 . . . . . . . . . . . . . . . . . 18 ([𝑣 / 𝑛]{𝑣, 𝑛} ∈ ran 𝐸𝑣 / 𝑛{𝑣, 𝑛} ∈ ran 𝐸)
25 csbprg 4191 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ∈ V → 𝑣 / 𝑛{𝑣, 𝑛} = {𝑣 / 𝑛𝑣, 𝑣 / 𝑛𝑛})
26 csbconstg 3512 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ V → 𝑣 / 𝑛𝑣 = 𝑣)
27 csbvarg 3955 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑣 ∈ V → 𝑣 / 𝑛𝑛 = 𝑣)
2826, 27preq12d 4220 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣 ∈ V → {𝑣 / 𝑛𝑣, 𝑣 / 𝑛𝑛} = {𝑣, 𝑣})
2925, 28eqtrd 2644 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ V → 𝑣 / 𝑛{𝑣, 𝑛} = {𝑣, 𝑣})
305, 29ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 𝑣 / 𝑛{𝑣, 𝑛} = {𝑣, 𝑣}
3130eleq1i 2679 . . . . . . . . . . . . . . . . . . 19 (𝑣 / 𝑛{𝑣, 𝑛} ∈ ran 𝐸 ↔ {𝑣, 𝑣} ∈ ran 𝐸)
32 equid 1926 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑣 = 𝑣
33 usgraedgrn 25910 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑉 USGrph 𝐸 ∧ {𝑣, 𝑣} ∈ ran 𝐸) → 𝑣𝑣)
34 eqneqall 2793 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑣 = 𝑣 → (𝑣𝑣𝐾 = 0))
3532, 33, 34mpsyl 66 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑉 USGrph 𝐸 ∧ {𝑣, 𝑣} ∈ ran 𝐸) → 𝐾 = 0)
3635ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 USGrph 𝐸 → ({𝑣, 𝑣} ∈ ran 𝐸𝐾 = 0))
3736adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → ({𝑣, 𝑣} ∈ ran 𝐸𝐾 = 0))
3837com12 32 . . . . . . . . . . . . . . . . . . . . 21 ({𝑣, 𝑣} ∈ ran 𝐸 → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0))
3938a1d 25 . . . . . . . . . . . . . . . . . . . 20 ({𝑣, 𝑣} ∈ ran 𝐸 → (𝑉 ∈ V → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0)))
40392a1d 26 . . . . . . . . . . . . . . . . . . 19 ({𝑣, 𝑣} ∈ ran 𝐸 → (𝑉 = {𝑣} → (𝐾 ∈ ℕ0 → (𝑉 ∈ V → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0)))))
4131, 40sylbi 206 . . . . . . . . . . . . . . . . . 18 (𝑣 / 𝑛{𝑣, 𝑛} ∈ ran 𝐸 → (𝑉 = {𝑣} → (𝐾 ∈ ℕ0 → (𝑉 ∈ V → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0)))))
4224, 41sylbi 206 . . . . . . . . . . . . . . . . 17 ([𝑣 / 𝑛]{𝑣, 𝑛} ∈ ran 𝐸 → (𝑉 = {𝑣} → (𝐾 ∈ ℕ0 → (𝑉 ∈ V → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0)))))
4322, 42syl 17 . . . . . . . . . . . . . . . 16 ((#‘{𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 1 → (𝑉 = {𝑣} → (𝐾 ∈ ℕ0 → (𝑉 ∈ V → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0)))))
4421, 43syl6bi 242 . . . . . . . . . . . . . . 15 (𝐾 = 1 → ((#‘{𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 → (𝑉 = {𝑣} → (𝐾 ∈ ℕ0 → (𝑉 ∈ V → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0))))))
4520, 44jaoi 393 . . . . . . . . . . . . . 14 ((𝐾 = 0 ∨ 𝐾 = 1) → ((#‘{𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 → (𝑉 = {𝑣} → (𝐾 ∈ ℕ0 → (𝑉 ∈ V → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0))))))
4617, 45mpcom 37 . . . . . . . . . . . . 13 ((#‘{𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 → (𝑉 = {𝑣} → (𝐾 ∈ ℕ0 → (𝑉 ∈ V → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0)))))
4746com12 32 . . . . . . . . . . . 12 (𝑉 = {𝑣} → ((#‘{𝑛 ∈ {𝑣} ∣ {𝑣, 𝑛} ∈ ran 𝐸}) = 𝐾 → (𝐾 ∈ ℕ0 → (𝑉 ∈ V → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0)))))
4816, 47sylbid 229 . . . . . . . . . . 11 (𝑉 = {𝑣} → (∀𝑘𝑉 (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾 → (𝐾 ∈ ℕ0 → (𝑉 ∈ V → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0)))))
4948com24 93 . . . . . . . . . 10 (𝑉 = {𝑣} → (𝑉 ∈ V → (𝐾 ∈ ℕ0 → (∀𝑘𝑉 (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾 → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0)))))
5049exlimiv 1845 . . . . . . . . 9 (∃𝑣 𝑉 = {𝑣} → (𝑉 ∈ V → (𝐾 ∈ ℕ0 → (∀𝑘𝑉 (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾 → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0)))))
5150com12 32 . . . . . . . 8 (𝑉 ∈ V → (∃𝑣 𝑉 = {𝑣} → (𝐾 ∈ ℕ0 → (∀𝑘𝑉 (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾 → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0)))))
523, 51sylbid 229 . . . . . . 7 (𝑉 ∈ V → ((#‘𝑉) = 1 → (𝐾 ∈ ℕ0 → (∀𝑘𝑉 (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾 → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → 𝐾 = 0)))))
5352com25 97 . . . . . 6 (𝑉 ∈ V → ((𝐸 ∈ V ∧ 𝑉 USGrph 𝐸) → (𝐾 ∈ ℕ0 → (∀𝑘𝑉 (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾 → ((#‘𝑉) = 1 → 𝐾 = 0)))))
5453expdimp 452 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 USGrph 𝐸 → (𝐾 ∈ ℕ0 → (∀𝑘𝑉 (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾 → ((#‘𝑉) = 1 → 𝐾 = 0)))))
552, 54mpcom 37 . . . 4 (𝑉 USGrph 𝐸 → (𝐾 ∈ ℕ0 → (∀𝑘𝑉 (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾 → ((#‘𝑉) = 1 → 𝐾 = 0))))
56553imp 1249 . . 3 ((𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑘𝑉 (#‘{𝑛𝑉 ∣ {𝑘, 𝑛} ∈ ran 𝐸}) = 𝐾) → ((#‘𝑉) = 1 → 𝐾 = 0))
571, 56syl 17 . 2 (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ((#‘𝑉) = 1 → 𝐾 = 0))
5857impcom 445 1 (((#‘𝑉) = 1 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → 𝐾 = 0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173  [wsbc 3402  ⦋csb 3499  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ran crn 5039  ‘cfv 5804  0cc0 9815  1c1 9816  ℕ0cn0 11169  #chash 12979   USGrph cusg 25859   RegUSGrph crusgra 26450 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-usgra 25862  df-nbgra 25949  df-vdgr 26421  df-rgra 26451  df-rusgra 26452 This theorem is referenced by:  frgrareg  26644
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