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Theorem konigsberg 16648

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 16636 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 16647	. . . 4 ⊢ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})
5		elpri 3728	. . . . 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 9374	. . . . . . . 8 ⊢ 0 < 2
7		0re 8316	. . . . . . . . 9 ⊢ 0 ∈ ℝ
8		2re 9353	. . . . . . . . 9 ⊢ 2 ∈ ℝ
9	7, 8	ltnsymi 8415	. . . . . . . 8 ⊢ (0 < 2 → ¬ 2 < 0)
10	6, 9	ax-mp 5	. . . . . . 7 ⊢ ¬ 2 < 0
11		breq2 4129	. . . . . . 7 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → (2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 0))
12	10, 11	mtbiri 686	. . . . . 6 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 0 → ¬ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}))
13	8	ltnri 8408	. . . . . . 7 ⊢ ¬ 2 < 2
14		breq2 4129	. . . . . . 7 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2 → (2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ↔ 2 < 2))
15	13, 14	mtbiri 686	. . . . . 6 ⊢ ((♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) = 2 → ¬ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}))
16	12, 15	jaoi 728	. . . . 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 649	. . 3 ⊢ ¬ (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2}
19	1, 2, 3	konigsbergumgr 16642	. . . 4 ⊢ 𝐺 ∈ UMGraph
20		0z 9634	. . . . . 6 ⊢ 0 ∈ ℤ
21		3z 9652	. . . . . 6 ⊢ 3 ∈ ℤ
22		fzfig 10845	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 3 ∈ ℤ) → (0...3) ∈ Fin)
23	20, 21, 22	mp2an 430	. . . . 5 ⊢ (0...3) ∈ Fin
24	1, 23	eqeltri 2311	. . . 4 ⊢ 𝑉 ∈ Fin
25	3	fveq2i 5693	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘⟨𝑉, 𝐸⟩)
26	24	elexi 2834	. . . . . . 7 ⊢ 𝑉 ∈ V
27		0nn0 9557	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
28		1nn0 9558	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
29		prexg 4344	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℕ₀ ∧ 1 ∈ ℕ₀) → {0, 1} ∈ V)
30	27, 28, 29	mp2an 430	. . . . . . . . . . 11 ⊢ {0, 1} ∈ V
31	30	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → {0, 1} ∈ V)
32		2nn0 9559	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
33		prexg 4344	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℕ₀ ∧ 2 ∈ ℕ₀) → {0, 2} ∈ V)
34	27, 32, 33	mp2an 430	. . . . . . . . . . 11 ⊢ {0, 2} ∈ V
35	34	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → {0, 2} ∈ V)
36		3nn0 9560	. . . . . . . . . . . 12 ⊢ 3 ∈ ℕ₀
37		prexg 4344	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℕ₀ ∧ 3 ∈ ℕ₀) → {0, 3} ∈ V)
38	27, 36, 37	mp2an 430	. . . . . . . . . . 11 ⊢ {0, 3} ∈ V
39	38	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → {0, 3} ∈ V)
40		prexg 4344	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℕ₀ ∧ 2 ∈ ℕ₀) → {1, 2} ∈ V)
41	28, 32, 40	mp2an 430	. . . . . . . . . . 11 ⊢ {1, 2} ∈ V
42	41	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → {1, 2} ∈ V)
43		prexg 4344	. . . . . . . . . . . 12 ⊢ ((2 ∈ ℕ₀ ∧ 3 ∈ ℕ₀) → {2, 3} ∈ V)
44	32, 36, 43	mp2an 430	. . . . . . . . . . 11 ⊢ {2, 3} ∈ V
45	44	a1i 9	. . . . . . . . . 10 ⊢ (⊤ → {2, 3} ∈ V)
46	31, 35, 39, 42, 42, 45, 45	s7cld 11533	. . . . . . . . 9 ⊢ (⊤ → ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V)
47	46	mptru 1411	. . . . . . . 8 ⊢ ⟨“{0, 1} {0, 2} {0, 3} {1, 2} {1, 2} {2, 3} {2, 3}”⟩ ∈ Word V
48	2, 47	eqeltri 2311	. . . . . . 7 ⊢ 𝐸 ∈ Word V
49		opvtxfv 16177	. . . . . . 7 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ Word V) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
50	26, 48, 49	mp2an 430	. . . . . 6 ⊢ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉
51	25, 50	eqtr2i 2260	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
52	51	eulerpathum 16636	. . . 4 ⊢ ((𝐺 ∈ UMGraph ∧ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) ∧ 𝑉 ∈ Fin) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})
53	19, 24, 52	mp3an13 1369	. . 3 ⊢ (∃𝑗 𝑗 ∈ (EulerPaths‘𝐺) → (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)}) ∈ {0, 2})
54	18, 53	mto 672	. 2 ⊢ ¬ ∃𝑗 𝑗 ∈ (EulerPaths‘𝐺)
55		notm0 3542	. 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 720 = wceq 1402 ⊤wtru 1403 ∃wex 1545 ∈ wcel 2209 {crab 2532 Vcvv 2821 ∅c0 3520 {cpr 3706 ⟨cop 3708 class class class wbr 4125 ‘cfv 5372 (class class class)co 6075 Fincfn 7012 0cc0 8169 1c1 8170 < clt 8350 2c2 9334 3c3 9335 ℕ₀cn0 9542 ℤcz 9623 ...cfz 10390 ♯chash 11192 Word cword 11282 ⟨“cs7 11504 ∥ cdvds 12532 Vtxcvtx 16167 UMGraphcumgr 16247 VtxDegcvtxdg 16441 EulerPathsceupth 16597

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288

This theorem depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-ifp 991 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-xor 1425 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-tp 3713 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-frec 6652 df-1o 6677 df-2o 6678 df-oadd 6681 df-er 6797 df-map 6914 df-en 7013 df-dom 7014 df-fin 7015 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-5 9345 df-6 9346 df-7 9347 df-8 9348 df-9 9349 df-n0 9543 df-z 9624 df-dec 9757 df-uz 9901 df-q 9999 df-rp 10034 df-xadd 10154 df-fz 10391 df-fzo 10528 df-fl 10683 df-mod 10738 df-seqfrec 10863 df-exp 10954 df-ihash 11193 df-word 11283 df-concat 11337 df-s1 11362 df-s2 11506 df-s3 11507 df-s4 11508 df-s5 11509 df-s6 11510 df-s7 11511 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-dvds 12533 df-ndx 13333 df-slot 13334 df-base 13336 df-edgf 16160 df-vtx 16169 df-iedg 16170 df-edg 16213 df-uhgrm 16224 df-ushgrm 16225 df-upgren 16248 df-umgren 16249 df-uspgren 16310 df-subgr 16409 df-vtxdg 16442 df-wlks 16473 df-trls 16536 df-eupth 16598

This theorem is referenced by: (None)

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