Theorem konigsberg 30201

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 30185 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.

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 30200	. . 3 ⊢ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})
5		elpri 4601	. . . 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 12231	. . . . . . 7 ⊢ 0 < 2
7		0re 11117	. . . . . . . 8 ⊢ 0 ∈ ℝ
8		2re 12202	. . . . . . . 8 ⊢ 2 ∈ ℝ
9	7, 8	ltnsymi 11235	. . . . . . 7 ⊢ (0 < 2 → ¬ 2 < 0)
10	6, 9	ax-mp 5	. . . . . 6 ⊢ ¬ 2 < 0
11		breq2 5096	. . . . . 6 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → (2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 0))
12	10, 11	mtbiri 327	. . . . 5 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → ¬ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}))
13	8	ltnri 11225	. . . . . 6 ⊢ ¬ 2 < 2
14		breq2 5096	. . . . . 6 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2 → (2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 2))
15	13, 14	mtbiri 327	. . . . 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 200	. 2 ⊢ ¬ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}
19	1, 2, 3	konigsbergumgr 30195	. . . . 5 ⊢ 𝐺 ∈ UMGraph
20		umgrupgr 29048	. . . . 5 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph)
21	19, 20	ax-mp 5	. . . 4 ⊢ 𝐺 ∈ UPGraph
22	3	fveq2i 6825	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘⟨𝑉, 𝐸⟩)
23	1	ovexi 7383	. . . . . . 7 ⊢ 𝑉 ∈ V
24		s7cli 14792	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
25	2, 24	eqeltri 2824	. . . . . . 7 ⊢ 𝐸 ∈ Word V
26		opvtxfv 28949	. . . . . . 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 2753	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
29	28	eulerpath 30185	. . . 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 2953	. 2 ⊢ (¬ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} → (EulerPaths‘𝐺) = ∅)
32	18, 31	ax-mp 5	1 ⊢ (EulerPaths‘𝐺) = ∅

Syntax hints:  ¬ wn 3   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {crab 3394  Vcvv 3436  ∅c0 4284  {cpr 4579  ⟨cop 4583   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349  0cc0 11009  1c1 11010   < clt 11149  2c2 12183  3c3 12184  ...cfz 13410  ♯chash 14237  Word cword 14420  ⟨“cs7 14753   ∥ cdvds 16163  Vtxcvtx 28941  UPGraphcupgr 29025  UMGraphcumgr 29026  VtxDegcvtxdg 29411  EulerPathsceupth 30141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-er 8625  df-map 8755  df-pm 8756  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-inf 9333  df-dju 9797  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-n0 12385  df-xnn0 12458  df-z 12472  df-uz 12736  df-rp 12894  df-xadd 13015  df-fz 13411  df-fzo 13558  df-seq 13909  df-exp 13969  df-hash 14238  df-word 14421  df-concat 14478  df-s1 14503  df-s2 14755  df-s3 14756  df-s4 14757  df-s5 14758  df-s6 14759  df-s7 14760  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-dvds 16164  df-vtx 28943  df-iedg 28944  df-edg 28993  df-uhgr 29003  df-ushgr 29004  df-upgr 29027  df-umgr 29028  df-uspgr 29095  df-vtxdg 29412  df-wlks 29545  df-trls 29636  df-eupth 30142

This theorem is referenced by: (None)