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Theorem konigsberg 27235
 Description: The Königsberg Bridge problem. If 𝐺 is the Königsberg graph, i.e. a graph on four vertices 0, 1, 2, 3, with edges {0, 1}, {0, 2}, {0, 3}, {1, 2}, {1, 2}, {2, 3}, {2, 3}, then vertices 0, 1, 3 each have degree three, and 2 has degree five, so there are four vertices of odd degree and thus by eulerpath 27219 the graph cannot have an Eulerian path. It is sufficient to show that there are 3 vertices of odd degree, since a graph having an Eulerian path can only have 0 or 2 vertices of odd degree. This is Metamath 100 proof #54. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 9-Mar-2021.)
Hypotheses
Ref Expression
konigsberg.v 𝑉 = (0...3)
konigsberg.e 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
konigsberg.g 𝐺 = ⟨𝑉, 𝐸
Assertion
Ref Expression
konigsberg (EulerPaths‘𝐺) = ∅

Proof of Theorem konigsberg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 konigsberg.v . . . 4 𝑉 = (0...3)
2 konigsberg.e . . . 4 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
3 konigsberg.g . . . 4 𝐺 = ⟨𝑉, 𝐸
41, 2, 3konigsberglem5 27234 . . 3 2 < (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})
5 elpri 4230 . . . 4 ((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} → ((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 ∨ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2))
6 2pos 11150 . . . . . . 7 0 < 2
7 0re 10078 . . . . . . . 8 0 ∈ ℝ
8 2re 11128 . . . . . . . 8 2 ∈ ℝ
97, 8ltnsymi 10194 . . . . . . 7 (0 < 2 → ¬ 2 < 0)
106, 9ax-mp 5 . . . . . 6 ¬ 2 < 0
11 breq2 4689 . . . . . 6 ((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → (2 < (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 0))
1210, 11mtbiri 316 . . . . 5 ((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → ¬ 2 < (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}))
138ltnri 10184 . . . . . 6 ¬ 2 < 2
14 breq2 4689 . . . . . 6 ((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2 → (2 < (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 2))
1513, 14mtbiri 316 . . . . 5 ((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2 → ¬ 2 < (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}))
1612, 15jaoi 393 . . . 4 (((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 ∨ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2) → ¬ 2 < (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}))
175, 16syl 17 . . 3 ((#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} → ¬ 2 < (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}))
184, 17mt2 191 . 2 ¬ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}
191, 2, 3konigsbergumgr 27229 . . . . 5 𝐺 ∈ UMGraph
20 umgrupgr 26043 . . . . 5 (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph)
2119, 20ax-mp 5 . . . 4 𝐺 ∈ UPGraph
223fveq2i 6232 . . . . . 6 (Vtx‘𝐺) = (Vtx‘⟨𝑉, 𝐸⟩)
23 ovex 6718 . . . . . . . 8 (0...3) ∈ V
241, 23eqeltri 2726 . . . . . . 7 𝑉 ∈ V
25 s7cli 13676 . . . . . . . 8 ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
262, 25eqeltri 2726 . . . . . . 7 𝐸 ∈ Word V
27 opvtxfv 25929 . . . . . . 7 ((𝑉 ∈ V ∧ 𝐸 ∈ Word V) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
2824, 26, 27mp2an 708 . . . . . 6 (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉
2922, 28eqtr2i 2674 . . . . 5 𝑉 = (Vtx‘𝐺)
3029eulerpath 27219 . . . 4 ((𝐺 ∈ UPGraph ∧ (EulerPaths‘𝐺) ≠ ∅) → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})
3121, 30mpan 706 . . 3 ((EulerPaths‘𝐺) ≠ ∅ → (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})
3231necon1bi 2851 . 2 (¬ (#‘{𝑥𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} → (EulerPaths‘𝐺) = ∅)
3318, 32ax-mp 5 1 (EulerPaths‘𝐺) = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 382   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  {crab 2945  Vcvv 3231  ∅c0 3948  {cpr 4212  ⟨cop 4216   class class class wbr 4685  ‘cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   < clt 10112  2c2 11108  3c3 11109  ...cfz 12364  #chash 13157  Word cword 13323  ⟨“cs7 13637   ∥ cdvds 15027  Vtxcvtx 25919  UPGraphcupgr 26020  UMGraphcumgr 26021  VtxDegcvtxdg 26417  EulerPathsceupth 27175 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-rp 11871  df-xadd 11985  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-s4 13641  df-s5 13642  df-s6 13643  df-s7 13644  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-dvds 15028  df-vtx 25921  df-iedg 25922  df-edg 25985  df-uhgr 25998  df-ushgr 25999  df-upgr 26022  df-umgr 26023  df-uspgr 26090  df-vtxdg 26418  df-wlks 26551  df-trls 26645  df-eupth 27176 This theorem is referenced by: (None)
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