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Mirrors > Home > MPE Home > Th. List > konigsberg | Structured version Visualization version GIF version |
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 28296 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.) |
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〉 |
Ref | Expression |
---|---|
konigsberg | ⊢ (EulerPaths‘𝐺) = ∅ |
Step | Hyp | Ref | 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〉 | |
4 | 1, 2, 3 | konigsberglem5 28311 | . . 3 ⊢ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) |
5 | elpri 4553 | . . . 4 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} → ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 ∨ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2)) | |
6 | 2pos 11916 | . . . . . . 7 ⊢ 0 < 2 | |
7 | 0re 10818 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
8 | 2re 11887 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
9 | 7, 8 | ltnsymi 10934 | . . . . . . 7 ⊢ (0 < 2 → ¬ 2 < 0) |
10 | 6, 9 | ax-mp 5 | . . . . . 6 ⊢ ¬ 2 < 0 |
11 | breq2 5047 | . . . . . 6 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → (2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 0)) | |
12 | 10, 11 | mtbiri 330 | . . . . 5 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → ¬ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
13 | 8 | ltnri 10924 | . . . . . 6 ⊢ ¬ 2 < 2 |
14 | breq2 5047 | . . . . . 6 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2 → (2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 2)) | |
15 | 13, 14 | mtbiri 330 | . . . . 5 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2 → ¬ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
16 | 12, 15 | jaoi 857 | . . . 4 ⊢ (((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 ∨ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2) → ¬ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
17 | 5, 16 | syl 17 | . . 3 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} → ¬ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
18 | 4, 17 | mt2 203 | . 2 ⊢ ¬ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} |
19 | 1, 2, 3 | konigsbergumgr 28306 | . . . . 5 ⊢ 𝐺 ∈ UMGraph |
20 | umgrupgr 27166 | . . . . 5 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph) | |
21 | 19, 20 | ax-mp 5 | . . . 4 ⊢ 𝐺 ∈ UPGraph |
22 | 3 | fveq2i 6709 | . . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘〈𝑉, 𝐸〉) |
23 | 1 | ovexi 7236 | . . . . . . 7 ⊢ 𝑉 ∈ V |
24 | s7cli 14433 | . . . . . . . 8 ⊢ 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 ∈ Word V | |
25 | 2, 24 | eqeltri 2830 | . . . . . . 7 ⊢ 𝐸 ∈ Word V |
26 | opvtxfv 27067 | . . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ Word V) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
27 | 23, 25, 26 | mp2an 692 | . . . . . 6 ⊢ (Vtx‘〈𝑉, 𝐸〉) = 𝑉 |
28 | 22, 27 | eqtr2i 2763 | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) |
29 | 28 | eulerpath 28296 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (EulerPaths‘𝐺) ≠ ∅) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) |
30 | 21, 29 | mpan 690 | . . 3 ⊢ ((EulerPaths‘𝐺) ≠ ∅ → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) |
31 | 30 | necon1bi 2963 | . 2 ⊢ (¬ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} → (EulerPaths‘𝐺) = ∅) |
32 | 18, 31 | ax-mp 5 | 1 ⊢ (EulerPaths‘𝐺) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 847 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 {crab 3058 Vcvv 3401 ∅c0 4227 {cpr 4533 〈cop 4537 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 0cc0 10712 1c1 10713 < clt 10850 2c2 11868 3c3 11869 ...cfz 13078 ♯chash 13879 Word cword 14052 〈“cs7 14394 ∥ cdvds 15796 Vtxcvtx 27059 UPGraphcupgr 27143 UMGraphcumgr 27144 VtxDegcvtxdg 27525 EulerPathsceupth 28252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-oadd 8195 df-er 8380 df-map 8499 df-pm 8500 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-sup 9047 df-inf 9048 df-dju 9500 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-n0 12074 df-xnn0 12146 df-z 12160 df-uz 12422 df-rp 12570 df-xadd 12688 df-fz 13079 df-fzo 13222 df-seq 13558 df-exp 13619 df-hash 13880 df-word 14053 df-concat 14109 df-s1 14136 df-s2 14396 df-s3 14397 df-s4 14398 df-s5 14399 df-s6 14400 df-s7 14401 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-dvds 15797 df-vtx 27061 df-iedg 27062 df-edg 27111 df-uhgr 27121 df-ushgr 27122 df-upgr 27145 df-umgr 27146 df-uspgr 27213 df-vtxdg 27526 df-wlks 27659 df-trls 27752 df-eupth 28253 |
This theorem is referenced by: (None) |
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