Theorem konigsberg 30345

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 30329 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 30344	. . 3 ⊢ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})
5		elpri 4592	. . . 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 12278	. . . . . . 7 ⊢ 0 < 2
7		0re 11140	. . . . . . . 8 ⊢ 0 ∈ ℝ
8		2re 12249	. . . . . . . 8 ⊢ 2 ∈ ℝ
9	7, 8	ltnsymi 11259	. . . . . . 7 ⊢ (0 < 2 → ¬ 2 < 0)
10	6, 9	ax-mp 5	. . . . . 6 ⊢ ¬ 2 < 0
11		breq2 5090	. . . . . 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 11249	. . . . . 6 ⊢ ¬ 2 < 2
14		breq2 5090	. . . . . 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 858	. . . 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 30339	. . . . 5 ⊢ 𝐺 ∈ UMGraph
20		umgrupgr 29189	. . . . 5 ⊢ (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph)
21	19, 20	ax-mp 5	. . . 4 ⊢ 𝐺 ∈ UPGraph
22	3	fveq2i 6838	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘⟨𝑉, 𝐸⟩)
23	1	ovexi 7395	. . . . . . 7 ⊢ 𝑉 ∈ V
24		s7cli 14841	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
25	2, 24	eqeltri 2833	. . . . . . 7 ⊢ 𝐸 ∈ Word V
26		opvtxfv 29090	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ Word V) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
27	23, 25, 26	mp2an 693	. . . . . 6 ⊢ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉
28	22, 27	eqtr2i 2761	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
29	28	eulerpath 30329	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (EulerPaths‘𝐺) ≠ ∅) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})
30	21, 29	mpan 691	. . 3 ⊢ ((EulerPaths‘𝐺) ≠ ∅ → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})
31	30	necon1bi 2961	. 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 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3390  Vcvv 3430  ∅c0 4274  {cpr 4570  ⟨cop 4574   class class class wbr 5086  ‘cfv 6493  (class class class)co 7361  0cc0 11032  1c1 11033   < clt 11173  2c2 12230  3c3 12231  ...cfz 13455  ♯chash 14286  Word cword 14469  ⟨“cs7 14802   ∥ cdvds 16215  Vtxcvtx 29082  UPGraphcupgr 29166  UMGraphcumgr 29167  VtxDegcvtxdg 29552  EulerPathsceupth 30285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109  ax-pre-sup 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-er 8637  df-map 8769  df-pm 8770  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-inf 9350  df-dju 9819  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-div 11802  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-n0 12432  df-xnn0 12505  df-z 12519  df-uz 12783  df-rp 12937  df-xadd 13058  df-fz 13456  df-fzo 13603  df-seq 13958  df-exp 14018  df-hash 14287  df-word 14470  df-concat 14527  df-s1 14553  df-s2 14804  df-s3 14805  df-s4 14806  df-s5 14807  df-s6 14808  df-s7 14809  df-cj 15055  df-re 15056  df-im 15057  df-sqrt 15191  df-abs 15192  df-dvds 16216  df-vtx 29084  df-iedg 29085  df-edg 29134  df-uhgr 29144  df-ushgr 29145  df-upgr 29168  df-umgr 29169  df-uspgr 29236  df-vtxdg 29553  df-wlks 29686  df-trls 29777  df-eupth 30286

This theorem is referenced by: (None)