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Theorem konigsberg 16734

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 16722 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 16733	. . . 4 ⊢ 2 < (♯‘{𝑥 ∈ 𝑉 ∣ ¬ 2 ∥ ((VtxDeg‘𝐺)‘𝑥)})
5		elpri 3732	. . . . 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 9395	. . . . . . . 8 ⊢ 0 < 2
7		0re 8326	. . . . . . . . 9 ⊢ 0 ∈ ℝ
8		2re 9374	. . . . . . . . 9 ⊢ 2 ∈ ℝ
9	7, 8	ltnsymi 8425	. . . . . . . 8 ⊢ (0 < 2 → ¬ 2 < 0)
10	6, 9	ax-mp 5	. . . . . . 7 ⊢ ¬ 2 < 0
11		breq2 4134	. . . . . . 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 8418	. . . . . . 7 ⊢ ¬ 2 < 2
14		breq2 4134	. . . . . . 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 16728	. . . 4 ⊢ 𝐺 ∈ UMGraph
20		0z 9655	. . . . . 6 ⊢ 0 ∈ ℤ
21		3z 9673	. . . . . 6 ⊢ 3 ∈ ℤ
22		fzfig 10867	. . . . . 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 5698	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘⟨𝑉, 𝐸⟩)
26	24	elexi 2834	. . . . . . 7 ⊢ 𝑉 ∈ V
27		0nn0 9578	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
28		1nn0 9579	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
29		prexg 4349	. . . . . . . . . . . 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 9580	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
33		prexg 4349	. . . . . . . . . . . 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 9581	. . . . . . . . . . . 12 ⊢ 3 ∈ ℕ₀
37		prexg 4349	. . . . . . . . . . . 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 4349	. . . . . . . . . . . 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 4349	. . . . . . . . . . . 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 11555	. . . . . . . . 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 16263	. . . . . . 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 16722	. . . 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

This proof depends on syntax axioms: ¬ wn 3 ∨ wo 720 = wceq 1402 ⊤wtru 1403 ∃wex 1545 ∈ wcel 2209 {crab 2532 Vcvv 2821 ∅c0 3520 {cpr 3710 ⟨cop 3712 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 Fincfn 7022 0cc0 8179 1c1 8180 < clt 8360 2c2 9355 3c3 9356 ℕ₀cn0 9563 ℤcz 9644 ...cfz 10411 ♯chash 11214 Word cword 11304 ⟨“cs7 11526 ∥ cdvds 12554 Vtxcvtx 16253 UMGraphcumgr 16333 VtxDegcvtxdg 16527 EulerPathsceupth 16683

This proof depends on 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 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298

This proof 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 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-tp 3717 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-irdg 6641 df-frec 6662 df-1o 6687 df-2o 6688 df-oadd 6691 df-er 6807 df-map 6924 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-3 9364 df-4 9365 df-5 9366 df-6 9367 df-7 9368 df-8 9369 df-9 9370 df-n0 9564 df-z 9645 df-dec 9778 df-uz 9922 df-q 10020 df-rp 10055 df-xadd 10175 df-fz 10412 df-fzo 10550 df-fl 10705 df-mod 10760 df-seqfrec 10885 df-exp 10976 df-ihash 11215 df-word 11305 df-concat 11359 df-s1 11384 df-s2 11528 df-s3 11529 df-s4 11530 df-s5 11531 df-s6 11532 df-s7 11533 df-cj 11607 df-re 11608 df-im 11609 df-rsqrt 11764 df-abs 11765 df-dvds 12555 df-ndx 13355 df-slot 13356 df-base 13358 df-edgf 16246 df-vtx 16255 df-iedg 16256 df-edg 16299 df-uhgrm 16310 df-ushgrm 16311 df-upgren 16334 df-umgren 16335 df-uspgren 16396 df-subgr 16495 df-vtxdg 16528 df-wlks 16559 df-trls 16622 df-eupth 16684

This theorem is used by: (None)

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