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Theorem konigsberg 16956

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 16944 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 16955	. . . 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 9404	. . . . . . . 8 ⊢ 0 < 2
7		0re 8332	. . . . . . . . 9 ⊢ 0 ∈ ℝ
8		2re 9383	. . . . . . . . 9 ⊢ 2 ∈ ℝ
9	7, 8	ltnsymi 8432	. . . . . . . 8 ⊢ (0 < 2 → ¬ 2 < 0)
10	6, 9	ax-mp 5	. . . . . . 7 ⊢ ¬ 2 < 0
11		breq2 4135	. . . . . . 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 8425	. . . . . . 7 ⊢ ¬ 2 < 2
14		breq2 4135	. . . . . . 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 16950	. . . 4 ⊢ 𝐺 ∈ UMGraph
20		0z 9664	. . . . . 6 ⊢ 0 ∈ ℤ
21		3z 9682	. . . . . 6 ⊢ 3 ∈ ℤ
22		fzfig 10887	. . . . . 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 5701	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘⟨𝑉, 𝐸⟩)
26	24	elexi 2834	. . . . . . 7 ⊢ 𝑉 ∈ V
27		0nn0 9589	. . . . . . . . . . . 12 ⊢ 0 ∈ ℕ₀
28		1nn0 9590	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ₀
29		prexg 4351	. . . . . . . . . . . 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 9591	. . . . . . . . . . . 12 ⊢ 2 ∈ ℕ₀
33		prexg 4351	. . . . . . . . . . . 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 9592	. . . . . . . . . . . 12 ⊢ 3 ∈ ℕ₀
37		prexg 4351	. . . . . . . . . . . 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 4351	. . . . . . . . . . . 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 4351	. . . . . . . . . . . 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 11576	. . . . . . . . 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 16485	. . . . . . 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 16944	. . . 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 4131 ‘cfv 5380 (class class class)co 6088 Fincfn 7028 0cc0 8185 1c1 8186 < clt 8366 2c2 9364 3c3 9365 ℕ₀cn0 9574 ℤcz 9653 ...cfz 10427 ♯chash 11235 Word cword 11325 ⟨“cs7 11547 ∥ cdvds 12578 Vtxcvtx 16475 UMGraphcumgr 16555 VtxDegcvtxdg 16749 EulerPathsceupth 16905

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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304

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 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6581 df-irdg 6646 df-frec 6667 df-1o 6692 df-2o 6693 df-oadd 6696 df-er 6812 df-map 6929 df-en 7029 df-dom 7030 df-fin 7031 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-5 9375 df-6 9376 df-7 9377 df-8 9378 df-9 9379 df-n0 9575 df-z 9654 df-dec 9788 df-uz 9937 df-q 10035 df-rp 10071 df-xadd 10191 df-fz 10428 df-fzo 10566 df-fl 10721 df-mod 10780 df-seqfrec 10905 df-exp 10996 df-ihash 11236 df-word 11326 df-concat 11380 df-s1 11405 df-s2 11549 df-s3 11550 df-s4 11551 df-s5 11552 df-s6 11553 df-s7 11554 df-cj 11628 df-re 11629 df-im 11630 df-rsqrt 11785 df-abs 11786 df-dvds 12579 df-ndx 13412 df-slot 13413 df-base 13415 df-edgf 16468 df-vtx 16477 df-iedg 16478 df-edg 16521 df-uhgrm 16532 df-ushgrm 16533 df-upgren 16556 df-umgren 16557 df-uspgren 16618 df-subgr 16717 df-vtxdg 16750 df-wlks 16781 df-trls 16844 df-eupth 16906

This theorem is used by: (None)

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