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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 27947 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 27962 | . . 3 ⊢ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) |
5 | elpri 4579 | . . . 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 11728 | . . . . . . 7 ⊢ 0 < 2 | |
7 | 0re 10631 | . . . . . . . 8 ⊢ 0 ∈ ℝ | |
8 | 2re 11699 | . . . . . . . 8 ⊢ 2 ∈ ℝ | |
9 | 7, 8 | ltnsymi 10747 | . . . . . . 7 ⊢ (0 < 2 → ¬ 2 < 0) |
10 | 6, 9 | ax-mp 5 | . . . . . 6 ⊢ ¬ 2 < 0 |
11 | breq2 5061 | . . . . . 6 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → (2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 0)) | |
12 | 10, 11 | mtbiri 328 | . . . . 5 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → ¬ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
13 | 8 | ltnri 10737 | . . . . . 6 ⊢ ¬ 2 < 2 |
14 | breq2 5061 | . . . . . 6 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2 → (2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 2)) | |
15 | 13, 14 | mtbiri 328 | . . . . 5 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2 → ¬ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})) |
16 | 12, 15 | jaoi 851 | . . . 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 201 | . 2 ⊢ ¬ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} |
19 | 1, 2, 3 | konigsbergumgr 27957 | . . . . 5 ⊢ 𝐺 ∈ UMGraph |
20 | umgrupgr 26815 | . . . . 5 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph) | |
21 | 19, 20 | ax-mp 5 | . . . 4 ⊢ 𝐺 ∈ UPGraph |
22 | 3 | fveq2i 6666 | . . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘〈𝑉, 𝐸〉) |
23 | 1 | ovexi 7179 | . . . . . . 7 ⊢ 𝑉 ∈ V |
24 | s7cli 14235 | . . . . . . . 8 ⊢ 〈“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”〉 ∈ Word V | |
25 | 2, 24 | eqeltri 2906 | . . . . . . 7 ⊢ 𝐸 ∈ Word V |
26 | opvtxfv 26716 | . . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ Word V) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
27 | 23, 25, 26 | mp2an 688 | . . . . . 6 ⊢ (Vtx‘〈𝑉, 𝐸〉) = 𝑉 |
28 | 22, 27 | eqtr2i 2842 | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) |
29 | 28 | eulerpath 27947 | . . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (EulerPaths‘𝐺) ≠ ∅) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) |
30 | 21, 29 | mpan 686 | . . 3 ⊢ ((EulerPaths‘𝐺) ≠ ∅ → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}) |
31 | 30 | necon1bi 3041 | . 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 841 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 {crab 3139 Vcvv 3492 ∅c0 4288 {cpr 4559 〈cop 4563 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 0cc0 10525 1c1 10526 < clt 10663 2c2 11680 3c3 11681 ...cfz 12880 ♯chash 13678 Word cword 13849 〈“cs7 14196 ∥ cdvds 15595 Vtxcvtx 26708 UPGraphcupgr 26792 UMGraphcumgr 26793 VtxDegcvtxdg 27174 EulerPathsceupth 27903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ifp 1055 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-rp 12378 df-xadd 12496 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-s2 14198 df-s3 14199 df-s4 14200 df-s5 14201 df-s6 14202 df-s7 14203 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-vtx 26710 df-iedg 26711 df-edg 26760 df-uhgr 26770 df-ushgr 26771 df-upgr 26794 df-umgr 26795 df-uspgr 26862 df-vtxdg 27175 df-wlks 27308 df-trls 27401 df-eupth 27904 |
This theorem is referenced by: (None) |
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