Theorem konigsberg 23620

Description: The Konigsberg Bridge problem. If <. V , E >. is the 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 eupath 23614 the graph cannot have an Eulerian path. This is Metamath 100 proof #54. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)

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 } ">

Assertion

Ref	Expression
konigsberg	|- ( V EulPaths E ) = (/)

Proof of Theorem konigsberg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prfi 7598	. . . . . 6 |- { 0 , 1 } e. Fin
2		3ne0 10428	. . . . . . 7 |- 3 =/= 0
3		1re 9397	. . . . . . . 8 |- 1 e. RR
4		1lt3 10502	. . . . . . . 8 |- 1 < 3
5	3, 4	gtneii 9498	. . . . . . 7 |- 3 =/= 1
6	2, 5	nelpri 3910	. . . . . 6 |- -. 3 e. { 0 , 1 }
7		3nn0 10609	. . . . . . 7 |- 3 e. NN0
8		hashunsng 12166	. . . . . . 7 |- ( 3 e. NN0 -> ( ( { 0 , 1 } e. Fin /\ -. 3 e. { 0 , 1 } ) -> ( # ` ( { 0 , 1 } u. { 3 } ) ) = ( ( # ` { 0 , 1 } ) + 1 ) ) )
9	7, 8	ax-mp 5	. . . . . 6 |- ( ( { 0 , 1 } e. Fin /\ -. 3 e. { 0 , 1 } ) -> ( # ` ( { 0 , 1 } u. { 3 } ) ) = ( ( # ` { 0 , 1 } ) + 1 ) )
10	1, 6, 9	mp2an 672	. . . . 5 |- ( # ` ( { 0 , 1 } u. { 3 } ) ) = ( ( # ` { 0 , 1 } ) + 1 )
11		df-3 10393	. . . . . 6 |- 3 = ( 2 + 1 )
12		0ne1 10401	. . . . . . . 8 |- 0 =/= 1
13		0nn0 10606	. . . . . . . . 9 |- 0 e. NN0
14		1nn0 10607	. . . . . . . . 9 |- 1 e. NN0
15		hashprg 12167	. . . . . . . . 9 |- ( ( 0 e. NN0 /\ 1 e. NN0 ) -> ( 0 =/= 1 <-> ( # ` { 0 , 1 } ) = 2 ) )
16	13, 14, 15	mp2an 672	. . . . . . . 8 |- ( 0 =/= 1 <-> ( # ` { 0 , 1 } ) = 2 )
17	12, 16	mpbi 208	. . . . . . 7 |- ( # ` { 0 , 1 } ) = 2
18	17	oveq1i 6113	. . . . . 6 |- ( ( # ` { 0 , 1 } ) + 1 ) = ( 2 + 1 )
19	11, 18	eqtr4i 2466	. . . . 5 |- 3 = ( ( # ` { 0 , 1 } ) + 1 )
20	10, 19	eqtr4i 2466	. . . 4 |- ( # ` ( { 0 , 1 } u. { 3 } ) ) = 3
21		konigsberg.v	. . . . . . . 8 |- V = ( 0 ... 3 )
22		fzfi 11806	. . . . . . . 8 |- ( 0 ... 3 ) e. Fin
23	21, 22	eqeltri 2513	. . . . . . 7 |- V e. Fin
24		ssrab2 3449	. . . . . . 7 |- { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } C_ V
25		ssfi 7545	. . . . . . 7 |- ( ( V e. Fin /\ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } C_ V ) -> { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } e. Fin )
26	23, 24, 25	mp2an 672	. . . . . 6 |- { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } e. Fin
27		nn0uz 10907	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
28	7, 27	eleqtri 2515	. . . . . . . . . . 11 |- 3 e. ( ZZ>= ` 0 )
29		eluzfz1 11470	. . . . . . . . . . 11 |- ( 3 e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... 3 ) )
30	28, 29	ax-mp 5	. . . . . . . . . 10 |- 0 e. ( 0 ... 3 )
31	30, 21	eleqtrri 2516	. . . . . . . . 9 |- 0 e. V
32		2nn 10491	. . . . . . . . . 10 |- 2 e. NN
33		1nn 10345	. . . . . . . . . 10 |- 1 e. NN
34		2t1e2 10482	. . . . . . . . . . . 12 |- ( 2 x. 1 ) = 2
35	34	oveq1i 6113	. . . . . . . . . . 11 |- ( ( 2 x. 1 ) + 1 ) = ( 2 + 1 )
36	35, 11	eqtr4i 2466	. . . . . . . . . 10 |- ( ( 2 x. 1 ) + 1 ) = 3
37		1lt2 10500	. . . . . . . . . 10 |- 1 < 2
38	32, 14, 33, 36, 37	ndvdsi 13626	. . . . . . . . 9 |- -. 2 || 3
39		fveq2 5703	. . . . . . . . . . . . 13 |- ( x = 0 -> ( ( V VDeg E ) ` x ) = ( ( V VDeg E ) ` 0 ) )
40	23	elexi 2994	. . . . . . . . . . . . . 14 |- V e. _V
41		df-s6 12491	. . . . . . . . . . . . . . 15 |- <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> = ( <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } "> concat <" { 2 , 3 } "> )
42		df-s5 12490	. . . . . . . . . . . . . . . 16 |- <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } "> = ( <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> concat <" { 1 , 2 } "> )
43		df-s4 12489	. . . . . . . . . . . . . . . . 17 |- <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> = ( <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> concat <" { 1 , 2 } "> )
44		df-s3 12488	. . . . . . . . . . . . . . . . . 18 |- <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> = ( <" { 0 , 1 } { 0 , 2 } "> concat <" { 0 , 3 } "> )
45		df-s2 12487	. . . . . . . . . . . . . . . . . . 19 |- <" { 0 , 1 } { 0 , 2 } "> = ( <" { 0 , 1 } "> concat <" { 0 , 2 } "> )
46		1le3 10550	. . . . . . . . . . . . . . . . . . . . . . 23 |- 1 <_ 3
47	14, 27	eleqtri 2515	. . . . . . . . . . . . . . . . . . . . . . . 24 |- 1 e. ( ZZ>= ` 0 )
48		3z 10691	. . . . . . . . . . . . . . . . . . . . . . . 24 |- 3 e. ZZ
49		elfz5 11457	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( 1 e. ( ZZ>= ` 0 ) /\ 3 e. ZZ ) -> ( 1 e. ( 0 ... 3 ) <-> 1 <_ 3 ) )
50	47, 48, 49	mp2an 672	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( 1 e. ( 0 ... 3 ) <-> 1 <_ 3 )
51	46, 50	mpbir 209	. . . . . . . . . . . . . . . . . . . . . 22 |- 1 e. ( 0 ... 3 )
52	51, 21	eleqtrri 2516	. . . . . . . . . . . . . . . . . . . . 21 |- 1 e. V
53	40, 31, 52	umgrabi 23616	. . . . . . . . . . . . . . . . . . . 20 |- ( T. -> { 0 , 1 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
54	53	s1cld 12306	. . . . . . . . . . . . . . . . . . 19 |- ( T. -> <" { 0 , 1 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
55		2re 10403	. . . . . . . . . . . . . . . . . . . . . . 23 |- 2 e. RR
56		3re 10407	. . . . . . . . . . . . . . . . . . . . . . 23 |- 3 e. RR
57		2lt3 10501	. . . . . . . . . . . . . . . . . . . . . . 23 |- 2 < 3
58	55, 56, 57	ltleii 9509	. . . . . . . . . . . . . . . . . . . . . 22 |- 2 <_ 3
59		2nn0 10608	. . . . . . . . . . . . . . . . . . . . . . . 24 |- 2 e. NN0
60	59, 27	eleqtri 2515	. . . . . . . . . . . . . . . . . . . . . . 23 |- 2 e. ( ZZ>= ` 0 )
61		elfz5 11457	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( 2 e. ( ZZ>= ` 0 ) /\ 3 e. ZZ ) -> ( 2 e. ( 0 ... 3 ) <-> 2 <_ 3 ) )
62	60, 48, 61	mp2an 672	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 2 e. ( 0 ... 3 ) <-> 2 <_ 3 )
63	58, 62	mpbir 209	. . . . . . . . . . . . . . . . . . . . 21 |- 2 e. ( 0 ... 3 )
64	63, 21	eleqtrri 2516	. . . . . . . . . . . . . . . . . . . 20 |- 2 e. V
65	40, 31, 64	umgrabi 23616	. . . . . . . . . . . . . . . . . . 19 |- ( T. -> { 0 , 2 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
66	45, 54, 65	cats1cld 12494	. . . . . . . . . . . . . . . . . 18 |- ( T. -> <" { 0 , 1 } { 0 , 2 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
67		eluzfz2 11471	. . . . . . . . . . . . . . . . . . . . 21 |- ( 3 e. ( ZZ>= ` 0 ) -> 3 e. ( 0 ... 3 ) )
68	28, 67	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 |- 3 e. ( 0 ... 3 )
69	68, 21	eleqtrri 2516	. . . . . . . . . . . . . . . . . . 19 |- 3 e. V
70	40, 31, 69	umgrabi 23616	. . . . . . . . . . . . . . . . . 18 |- ( T. -> { 0 , 3 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
71	44, 66, 70	cats1cld 12494	. . . . . . . . . . . . . . . . 17 |- ( T. -> <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
72	40, 52, 64	umgrabi 23616	. . . . . . . . . . . . . . . . 17 |- ( T. -> { 1 , 2 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
73	43, 71, 72	cats1cld 12494	. . . . . . . . . . . . . . . 16 |- ( T. -> <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
74	42, 73, 72	cats1cld 12494	. . . . . . . . . . . . . . 15 |- ( T. -> <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
75	40, 64, 69	umgrabi 23616	. . . . . . . . . . . . . . 15 |- ( T. -> { 2 , 3 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
76	41, 74, 75	cats1cld 12494	. . . . . . . . . . . . . 14 |- ( T. -> <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
77		wrd0 12264	. . . . . . . . . . . . . . . . . . . . 21 |- (/) e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 }
78	77	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( T. -> (/) e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
79	40, 31	vdeg0i 23615	. . . . . . . . . . . . . . . . . . . 20 |- ( ( V VDeg (/) ) ` 0 ) = 0
80		1e0p1 10795	. . . . . . . . . . . . . . . . . . . 20 |- 1 = ( 0 + 1 )
81		ax-1ne0 9363	. . . . . . . . . . . . . . . . . . . 20 |- 1 =/= 0
82		s0s1 12544	. . . . . . . . . . . . . . . . . . . 20 |- <" { 0 , 1 } "> = ( (/) concat <" { 0 , 1 } "> )
83	40, 78, 31, 79, 80, 52, 81, 82	vdegp1bi 23618	. . . . . . . . . . . . . . . . . . 19 |- ( ( V VDeg <" { 0 , 1 } "> ) ` 0 ) = 1
84		df-2 10392	. . . . . . . . . . . . . . . . . . 19 |- 2 = ( 1 + 1 )
85		2ne0 10426	. . . . . . . . . . . . . . . . . . 19 |- 2 =/= 0
86	40, 54, 31, 83, 84, 64, 85, 45	vdegp1bi 23618	. . . . . . . . . . . . . . . . . 18 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } "> ) ` 0 ) = 2
87	40, 66, 31, 86, 11, 69, 2, 44	vdegp1bi 23618	. . . . . . . . . . . . . . . . 17 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> ) ` 0 ) = 3
88	40, 71, 31, 87, 52, 81, 64, 85, 43	vdegp1ai 23617	. . . . . . . . . . . . . . . 16 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> ) ` 0 ) = 3
89	40, 73, 31, 88, 52, 81, 64, 85, 42	vdegp1ai 23617	. . . . . . . . . . . . . . 15 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } "> ) ` 0 ) = 3
90	40, 74, 31, 89, 64, 85, 69, 2, 41	vdegp1ai 23617	. . . . . . . . . . . . . 14 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> ) ` 0 ) = 3
91		konigsberg.e	. . . . . . . . . . . . . . 15 |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">
92		df-s7 12492	. . . . . . . . . . . . . . 15 |- <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } "> = ( <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> concat <" { 2 , 3 } "> )
93	91, 92	eqtri 2463	. . . . . . . . . . . . . 14 |- E = ( <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> concat <" { 2 , 3 } "> )
94	40, 76, 31, 90, 64, 85, 69, 2, 93	vdegp1ai 23617	. . . . . . . . . . . . 13 |- ( ( V VDeg E ) ` 0 ) = 3
95	39, 94	syl6eq 2491	. . . . . . . . . . . 12 |- ( x = 0 -> ( ( V VDeg E ) ` x ) = 3 )
96	95	breq2d 4316	. . . . . . . . . . 11 |- ( x = 0 -> ( 2 || ( ( V VDeg E ) ` x ) <-> 2 || 3 ) )
97	96	notbid 294	. . . . . . . . . 10 |- ( x = 0 -> ( -. 2 || ( ( V VDeg E ) ` x ) <-> -. 2 || 3 ) )
98	97	elrab 3129	. . . . . . . . 9 |- ( 0 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } <-> ( 0 e. V /\ -. 2 || 3 ) )
99	31, 38, 98	mpbir2an 911	. . . . . . . 8 |- 0 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
100		fveq2 5703	. . . . . . . . . . . . 13 |- ( x = 1 -> ( ( V VDeg E ) ` x ) = ( ( V VDeg E ) ` 1 ) )
101	40, 52	vdeg0i 23615	. . . . . . . . . . . . . . . . . . . 20 |- ( ( V VDeg (/) ) ` 1 ) = 0
102	40, 78, 52, 101, 80, 31, 12, 82	vdegp1ci 23619	. . . . . . . . . . . . . . . . . . 19 |- ( ( V VDeg <" { 0 , 1 } "> ) ` 1 ) = 1
103	3, 37	gtneii 9498	. . . . . . . . . . . . . . . . . . 19 |- 2 =/= 1
104	40, 54, 52, 102, 31, 12, 64, 103, 45	vdegp1ai 23617	. . . . . . . . . . . . . . . . . 18 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } "> ) ` 1 ) = 1
105	40, 66, 52, 104, 31, 12, 69, 5, 44	vdegp1ai 23617	. . . . . . . . . . . . . . . . 17 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> ) ` 1 ) = 1
106	40, 71, 52, 105, 84, 64, 103, 43	vdegp1bi 23618	. . . . . . . . . . . . . . . 16 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> ) ` 1 ) = 2
107	40, 73, 52, 106, 11, 64, 103, 42	vdegp1bi 23618	. . . . . . . . . . . . . . 15 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } "> ) ` 1 ) = 3
108	40, 74, 52, 107, 64, 103, 69, 5, 41	vdegp1ai 23617	. . . . . . . . . . . . . 14 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> ) ` 1 ) = 3
109	40, 76, 52, 108, 64, 103, 69, 5, 93	vdegp1ai 23617	. . . . . . . . . . . . 13 |- ( ( V VDeg E ) ` 1 ) = 3
110	100, 109	syl6eq 2491	. . . . . . . . . . . 12 |- ( x = 1 -> ( ( V VDeg E ) ` x ) = 3 )
111	110	breq2d 4316	. . . . . . . . . . 11 |- ( x = 1 -> ( 2 || ( ( V VDeg E ) ` x ) <-> 2 || 3 ) )
112	111	notbid 294	. . . . . . . . . 10 |- ( x = 1 -> ( -. 2 || ( ( V VDeg E ) ` x ) <-> -. 2 || 3 ) )
113	112	elrab 3129	. . . . . . . . 9 |- ( 1 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } <-> ( 1 e. V /\ -. 2 || 3 ) )
114	52, 38, 113	mpbir2an 911	. . . . . . . 8 |- 1 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
115		prssi 4041	. . . . . . . 8 |- ( ( 0 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } /\ 1 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) -> { 0 , 1 } C_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } )
116	99, 114, 115	mp2an 672	. . . . . . 7 |- { 0 , 1 } C_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
117		fveq2 5703	. . . . . . . . . . . . 13 |- ( x = 3 -> ( ( V VDeg E ) ` x ) = ( ( V VDeg E ) ` 3 ) )
118	40, 69	vdeg0i 23615	. . . . . . . . . . . . . . . . . . . 20 |- ( ( V VDeg (/) ) ` 3 ) = 0
119		0re 9398	. . . . . . . . . . . . . . . . . . . . 21 |- 0 e. RR
120		3pos 10427	. . . . . . . . . . . . . . . . . . . . 21 |- 0 < 3
121	119, 120	ltneii 9499	. . . . . . . . . . . . . . . . . . . 20 |- 0 =/= 3
122	3, 4	ltneii 9499	. . . . . . . . . . . . . . . . . . . 20 |- 1 =/= 3
123	40, 78, 69, 118, 31, 121, 52, 122, 82	vdegp1ai 23617	. . . . . . . . . . . . . . . . . . 19 |- ( ( V VDeg <" { 0 , 1 } "> ) ` 3 ) = 0
124	55, 57	ltneii 9499	. . . . . . . . . . . . . . . . . . 19 |- 2 =/= 3
125	40, 54, 69, 123, 31, 121, 64, 124, 45	vdegp1ai 23617	. . . . . . . . . . . . . . . . . 18 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } "> ) ` 3 ) = 0
126	40, 66, 69, 125, 80, 31, 121, 44	vdegp1ci 23619	. . . . . . . . . . . . . . . . 17 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> ) ` 3 ) = 1
127	40, 71, 69, 126, 52, 122, 64, 124, 43	vdegp1ai 23617	. . . . . . . . . . . . . . . 16 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> ) ` 3 ) = 1
128	40, 73, 69, 127, 52, 122, 64, 124, 42	vdegp1ai 23617	. . . . . . . . . . . . . . 15 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } "> ) ` 3 ) = 1
129	40, 74, 69, 128, 84, 64, 124, 41	vdegp1ci 23619	. . . . . . . . . . . . . 14 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> ) ` 3 ) = 2
130	40, 76, 69, 129, 11, 64, 124, 93	vdegp1ci 23619	. . . . . . . . . . . . 13 |- ( ( V VDeg E ) ` 3 ) = 3
131	117, 130	syl6eq 2491	. . . . . . . . . . . 12 |- ( x = 3 -> ( ( V VDeg E ) ` x ) = 3 )
132	131	breq2d 4316	. . . . . . . . . . 11 |- ( x = 3 -> ( 2 || ( ( V VDeg E ) ` x ) <-> 2 || 3 ) )
133	132	notbid 294	. . . . . . . . . 10 |- ( x = 3 -> ( -. 2 || ( ( V VDeg E ) ` x ) <-> -. 2 || 3 ) )
134	133	elrab 3129	. . . . . . . . 9 |- ( 3 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } <-> ( 3 e. V /\ -. 2 || 3 ) )
135	69, 38, 134	mpbir2an 911	. . . . . . . 8 |- 3 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
136		snssi 4029	. . . . . . . 8 |- ( 3 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } -> { 3 } C_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } )
137	135, 136	ax-mp 5	. . . . . . 7 |- { 3 } C_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
138	116, 137	unssi 3543	. . . . . 6 |- ( { 0 , 1 } u. { 3 } ) C_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
139		ssdomg 7367	. . . . . 6 |- ( { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } e. Fin -> ( ( { 0 , 1 } u. { 3 } ) C_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } -> ( { 0 , 1 } u. { 3 } ) ~<_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) )
140	26, 138, 139	mp2 9	. . . . 5 |- ( { 0 , 1 } u. { 3 } ) ~<_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
141		snfi 7402	. . . . . . 7 |- { 3 } e. Fin
142		unfi 7591	. . . . . . 7 |- ( ( { 0 , 1 } e. Fin /\ { 3 } e. Fin ) -> ( { 0 , 1 } u. { 3 } ) e. Fin )
143	1, 141, 142	mp2an 672	. . . . . 6 |- ( { 0 , 1 } u. { 3 } ) e. Fin
144		hashdom 12154	. . . . . 6 |- ( ( ( { 0 , 1 } u. { 3 } ) e. Fin /\ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } e. Fin ) -> ( ( # ` ( { 0 , 1 } u. { 3 } ) ) <_ ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) <-> ( { 0 , 1 } u. { 3 } ) ~<_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) )
145	143, 26, 144	mp2an 672	. . . . 5 |- ( ( # ` ( { 0 , 1 } u. { 3 } ) ) <_ ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) <-> ( { 0 , 1 } u. { 3 } ) ~<_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } )
146	140, 145	mpbir 209	. . . 4 |- ( # ` ( { 0 , 1 } u. { 3 } ) ) <_ ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } )
147	20, 146	eqbrtrri 4325	. . 3 |- 3 <_ ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } )
148		hashcl 12138	. . . . . 6 |- ( { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } e. Fin -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) e. NN0 )
149	26, 148	ax-mp 5	. . . . 5 |- ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) e. NN0
150	149	nn0rei 10602	. . . 4 |- ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) e. RR
151	56, 150	lenlti 9506	. . 3 |- ( 3 <_ ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) <-> -. ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) < 3 )
152	147, 151	mpbi 208	. 2 |- -. ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) < 3
153		eupath 23614	. . . 4 |- ( ( V EulPaths E ) =/= (/) -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) e. { 0 , 2 } )
154		elpri 3909	. . . 4 |- ( ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) e. { 0 , 2 } -> ( ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 0 \/ ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 2 ) )
155		id 22	. . . . . 6 |- ( ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 0 -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 0 )
156	155, 120	syl6eqbr 4341	. . . . 5 |- ( ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 0 -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) < 3 )
157		id 22	. . . . . 6 |- ( ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 2 -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 2 )
158	157, 57	syl6eqbr 4341	. . . . 5 |- ( ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 2 -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) < 3 )
159	156, 158	jaoi 379	. . . 4 |- ( ( ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 0 \/ ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 2 ) -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) < 3 )
160	153, 154, 159	3syl 20	. . 3 |- ( ( V EulPaths E ) =/= (/) -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) < 3 )
161	160	necon1bi 2666	. 2 |- ( -. ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) < 3 -> ( V EulPaths E ) = (/) )
162	152, 161	ax-mp 5	1 |- ( V EulPaths E ) = (/)

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1369   T. wtru 1370   e. wcel 1756   =/= wne 2618   {crab 2731   \ cdif 3337   u. cun 3338   C_ wss 3340   (/)c0 3649   ~Pcpw 3872   {csn 3889   {cpr 3891   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   ~<_ cdom 7320   Fincfn 7322   0cc0 9294   1c1 9295   + caddc 9297   x. cmul 9299   < clt 9430   <_ cle 9431   2c2 10383   3c3 10384   NN0cn0 10591   ZZcz 10658   ZZ>=cuz 10873   ...cfz 11449   #chash 12115  Word cword 12233   concat cconcat 12235   <"cs1 12236   <"cs2 12480   <"cs3 12481   <"cs4 12482   <"cs5 12483   <"cs6 12484   <"cs7 12485   || cdivides 13547   VDeg cvdg 23575   EulPaths ceup 23595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-xadd 11102  df-fz 11450  df-fzo 11561  df-seq 11819  df-exp 11878  df-hash 12116  df-word 12241  df-concat 12243  df-s1 12244  df-s2 12487  df-s3 12488  df-s4 12489  df-s5 12490  df-s6 12491  df-s7 12492  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-dvds 13548  df-umgra 23259  df-vdgr 23576  df-eupa 23596

This theorem is referenced by: (None)