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Theorem konigsberg 16614

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 16602 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 16613	. . . 4 ⊢ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})
5		elpri 3717	. . . . 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 9345	. . . . . . . 8 ⊢ 0 < 2
7		0re 8290	. . . . . . . . 9 ⊢ 0 ∈ ℝ
8		2re 9324	. . . . . . . . 9 ⊢ 2 ∈ ℝ
9	7, 8	ltnsymi 8389	. . . . . . . 8 ⊢ (0 < 2 → ¬ 2 < 0)
10	6, 9	ax-mp 5	. . . . . . 7 ⊢ ¬ 2 < 0
11		breq2 4118	. . . . . . 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 8382	. . . . . . 7 ⊢ ¬ 2 < 2
14		breq2 4118	. . . . . . 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 16608	. . . 4 ⊢ 𝐺 ∈ UMGraph
20		0z 9605	. . . . . 6 ⊢ 0 ∈ ℤ
21		3z 9623	. . . . . 6 ⊢ 3 ∈ ℤ
22		fzfig 10816	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 3 ∈ ℤ) → (0...3) ∈ Fin)
23	20, 21, 22	mp2an 426	. . . . 5 ⊢ (0...3) ∈ Fin
24	1, 23	eqeltri 2307	. . . 4 ⊢ 𝑉 ∈ Fin
25	3	fveq2i 5678	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘⟨𝑉, 𝐸⟩)
26	24	elexi 2828	. . . . . . 7 ⊢ 𝑉 ∈ V
27		0nn0 9528	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
28		1nn0 9529	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
29		prexg 4330	. . . . . . . . . . . 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 9530	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
33		prexg 4330	. . . . . . . . . . . 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 9531	. . . . . . . . . . . 12 ⊢ 3 ∈ ℕ₀
37		prexg 4330	. . . . . . . . . . . 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 4330	. . . . . . . . . . . 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 4330	. . . . . . . . . . . 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 11500	. . . . . . . . 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 2307	. . . . . . 7 ⊢ 𝐸 ∈ Word V
49		opvtxfv 16143	. . . . . . 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 2256	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
52	51	eulerpathum 16602	. . . 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 3533	. 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 2205 {crab 2526 Vcvv 2815 ∅c0 3512 {cpr 3695 ⟨cop 3697 class class class wbr 4114 ‘cfv 5357 (class class class)co 6058 Fincfn 6988 0cc0 8143 1c1 8144 < clt 8324 2c2 9305 3c3 9306 ℕ₀cn0 9513 ℤcz 9594 ...cfz 10361 ♯chash 11163 Word cword 11249 ⟨“cs7 11471 ∥ cdvds 12498 Vtxcvtx 16133 UMGraphcumgr 16213 VtxDegcvtxdg 16407 EulerPathsceupth 16563

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262

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 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-tp 3702 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-2o 6661 df-oadd 6664 df-er 6780 df-map 6897 df-en 6989 df-dom 6990 df-fin 6991 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-5 9316 df-6 9317 df-7 9318 df-8 9319 df-9 9320 df-n0 9514 df-z 9595 df-dec 9728 df-uz 9872 df-q 9970 df-rp 10005 df-xadd 10125 df-fz 10362 df-fzo 10499 df-fl 10654 df-mod 10709 df-seqfrec 10834 df-exp 10925 df-ihash 11164 df-word 11250 df-concat 11304 df-s1 11329 df-s2 11473 df-s3 11474 df-s4 11475 df-s5 11476 df-s6 11477 df-s7 11478 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-dvds 12499 df-ndx 13299 df-slot 13300 df-base 13302 df-edgf 16126 df-vtx 16135 df-iedg 16136 df-edg 16179 df-uhgrm 16190 df-ushgrm 16191 df-upgren 16214 df-umgren 16215 df-uspgren 16276 df-subgr 16375 df-vtxdg 16408 df-wlks 16439 df-trls 16502 df-eupth 16564

This theorem is referenced by: (None)

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