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Theorem konigsberg 16417

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 eulerpathum 16405 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 variables 𝑥 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		konigsberg.v	. . . . 5 ⊢ 𝑉 = (0...3)
2		konigsberg.e	. . . . 5 ⊢ 𝐸 = ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩
3		konigsberg.g	. . . . 5 ⊢ 𝐺 = ⟨𝑉, 𝐸⟩
4	1, 2, 3	konigsberglem5 16416	. . . 4 ⊢ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})
5		elpri 3696	. . . . 5 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} → ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 ∨ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2))
6		2pos 9276	. . . . . . . 8 ⊢ 0 < 2
7		0re 8222	. . . . . . . . 9 ⊢ 0 ∈ ℝ
8		2re 9255	. . . . . . . . 9 ⊢ 2 ∈ ℝ
9	7, 8	ltnsymi 8321	. . . . . . . 8 ⊢ (0 < 2 → ¬ 2 < 0)
10	6, 9	ax-mp 5	. . . . . . 7 ⊢ ¬ 2 < 0
11		breq2 4097	. . . . . . 7 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → (2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 0))
12	10, 11	mtbiri 682	. . . . . 6 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → ¬ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}))
13	8	ltnri 8314	. . . . . . 7 ⊢ ¬ 2 < 2
14		breq2 4097	. . . . . . 7 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2 → (2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 2))
15	13, 14	mtbiri 682	. . . . . 6 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2 → ¬ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}))
16	12, 15	jaoi 724	. . . . 5 ⊢ (((♯‘{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 14	. . . 4 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2} → ¬ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}))
18	4, 17	mt2 645	. . 3 ⊢ ¬ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}
19	1, 2, 3	konigsbergumgr 16411	. . . 4 ⊢ 𝐺 ∈ UMGraph
20		0z 9534	. . . . . 6 ⊢ 0 ∈ ℤ
21		3z 9552	. . . . . 6 ⊢ 3 ∈ ℤ
22		fzfig 10738	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 3 ∈ ℤ) → (0...3) ∈ Fin)
23	20, 21, 22	mp2an 426	. . . . 5 ⊢ (0...3) ∈ Fin
24	1, 23	eqeltri 2304	. . . 4 ⊢ 𝑉 ∈ Fin
25	3	fveq2i 5651	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘⟨𝑉, 𝐸⟩)
26	24	elexi 2816	. . . . . . 7 ⊢ 𝑉 ∈ V
27		0nn0 9459	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
28		1nn0 9460	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
29		prexg 4307	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → {0, 1} ∈ V)
30	27, 28, 29	mp2an 426	. . . . . . . . . . 11 ⊢ {0, 1} ∈ V
31	30	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → {0, 1} ∈ V)
32		2nn0 9461	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
33		prexg 4307	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℕ₀ ∧ 2 ∈ ℕ₀) → {0, 2} ∈ V)
34	27, 32, 33	mp2an 426	. . . . . . . . . . 11 ⊢ {0, 2} ∈ V
35	34	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → {0, 2} ∈ V)
36		3nn0 9462	. . . . . . . . . . . 12 ⊢ 3 ∈ ℕ₀
37		prexg 4307	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℕ₀ ∧ 3 ∈ ℕ₀) → {0, 3} ∈ V)
38	27, 36, 37	mp2an 426	. . . . . . . . . . 11 ⊢ {0, 3} ∈ V
39	38	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → {0, 3} ∈ V)
40		prexg 4307	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℕ₀ ∧ 2 ∈ ℕ₀) → {1, 2} ∈ V)
41	28, 32, 40	mp2an 426	. . . . . . . . . . 11 ⊢ {1, 2} ∈ V
42	41	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → {1, 2} ∈ V)
43		prexg 4307	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℕ₀ ∧ 3 ∈ ℕ₀) → {2, 3} ∈ V)
44	32, 36, 43	mp2an 426	. . . . . . . . . . 11 ⊢ {2, 3} ∈ V
45	44	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → {2, 3} ∈ V)
46	31, 35, 39, 42, 42, 45, 45	s7cld 11413	. . . . . . . . 9 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V)
47	46	mptru 1407	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
48	2, 47	eqeltri 2304	. . . . . . 7 ⊢ 𝐸 ∈ Word V
49		opvtxfv 15946	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ Word V) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
50	26, 48, 49	mp2an 426	. . . . . 6 ⊢ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉
51	25, 50	eqtr2i 2253	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
52	51	eulerpathum 16405	. . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})
53	19, 24, 52	mp3an13 1365	. . 3 ⊢ (∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})
54	18, 53	mto 668	. 2 ⊢ ¬ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺)
55		notm0 3517	. 2 ⊢ (¬ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ↔ (EulerPaths‘𝐺) = ∅)
56	54, 55	mpbi 145	1 ⊢ (EulerPaths‘𝐺) = ∅

Colors of variables: wff set class

Syntax hints: ¬ wn 3 ∨ wo 716 = wceq 1398 ⊤wtru 1399 ∃wex 1541 ∈ wcel 2202 {crab 2515 Vcvv 2803 ∅c0 3496 {cpr 3674 ⟨cop 3676 class class class wbr 4093 ‘cfv 5333 (class class class)co 6028 Fincfn 6952 0cc0 8075 1c1 8076 < clt 8256 2c2 9236 3c3 9237 ℕ₀cn0 9444 ℤcz 9523 ...cfz 10288 ♯chash 11083 Word cword 11162 ⟨“cs7 11384 ∥ cdvds 12411 Vtxcvtx 15936 UMGraphcumgr 16016 VtxDegcvtxdg 16210 EulerPathsceupth 16366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-ifp 987 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-tp 3681 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-frec 6600 df-1o 6625 df-2o 6626 df-oadd 6629 df-er 6745 df-map 6862 df-en 6953 df-dom 6954 df-fin 6955 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-reap 8797 df-ap 8804 df-div 8895 df-inn 9186 df-2 9244 df-3 9245 df-4 9246 df-5 9247 df-6 9248 df-7 9249 df-8 9250 df-9 9251 df-n0 9445 df-z 9524 df-dec 9656 df-uz 9800 df-q 9898 df-rp 9933 df-xadd 10052 df-fz 10289 df-fzo 10423 df-fl 10576 df-mod 10631 df-seqfrec 10756 df-exp 10847 df-ihash 11084 df-word 11163 df-concat 11217 df-s1 11242 df-s2 11386 df-s3 11387 df-s4 11388 df-s5 11389 df-s6 11390 df-s7 11391 df-cj 11465 df-re 11466 df-im 11467 df-rsqrt 11621 df-abs 11622 df-dvds 12412 df-ndx 13148 df-slot 13149 df-base 13151 df-edgf 15929 df-vtx 15938 df-iedg 15939 df-edg 15982 df-uhgrm 15993 df-ushgrm 15994 df-upgren 16017 df-umgren 16018 df-uspgren 16079 df-subgr 16178 df-vtxdg 16211 df-wlks 16242 df-trls 16305 df-eupth 16367

This theorem is referenced by: (None)

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