Theorem konigsberg 25771

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 25765 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 7877	. . . . . 6 |- { 0 , 1 } e. Fin
2		3ne0 10737	. . . . . . 7 |- 3 =/= 0
3		1re 9673	. . . . . . . 8 |- 1 e. RR
4		1lt3 10812	. . . . . . . 8 |- 1 < 3
5	3, 4	gtneii 9777	. . . . . . 7 |- 3 =/= 1
6	2, 5	nelpri 4000	. . . . . 6 |- -. 3 e. { 0 , 1 }
7		3nn0 10921	. . . . . . 7 |- 3 e. NN0
8		hashunsng 12609	. . . . . . 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 683	. . . . 5 |- ( # ` ( { 0 , 1 } u. { 3 } ) ) = ( ( # ` { 0 , 1 } ) + 1 )
11		df-3 10702	. . . . . 6 |- 3 = ( 2 + 1 )
12		0ne1 10710	. . . . . . . 8 |- 0 =/= 1
13		0nn0 10918	. . . . . . . . 9 |- 0 e. NN0
14		1nn0 10919	. . . . . . . . 9 |- 1 e. NN0
15		hashprg 12610	. . . . . . . . 9 |- ( ( 0 e. NN0 /\ 1 e. NN0 ) -> ( 0 =/= 1 <-> ( # ` { 0 , 1 } ) = 2 ) )
16	13, 14, 15	mp2an 683	. . . . . . . 8 |- ( 0 =/= 1 <-> ( # ` { 0 , 1 } ) = 2 )
17	12, 16	mpbi 213	. . . . . . 7 |- ( # ` { 0 , 1 } ) = 2
18	17	oveq1i 6330	. . . . . 6 |- ( ( # ` { 0 , 1 } ) + 1 ) = ( 2 + 1 )
19	11, 18	eqtr4i 2487	. . . . 5 |- 3 = ( ( # ` { 0 , 1 } ) + 1 )
20	10, 19	eqtr4i 2487	. . . 4 |- ( # ` ( { 0 , 1 } u. { 3 } ) ) = 3
21		konigsberg.v	. . . . . . . 8 |- V = ( 0 ... 3 )
22		fzfi 12223	. . . . . . . 8 |- ( 0 ... 3 ) e. Fin
23	21, 22	eqeltri 2536	. . . . . . 7 |- V e. Fin
24		ssrab2 3526	. . . . . . 7 |- { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } C_ V
25		ssfi 7823	. . . . . . 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 683	. . . . . 6 |- { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } e. Fin
27		nn0uz 11227	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
28	7, 27	eleqtri 2538	. . . . . . . . . . 11 |- 3 e. ( ZZ>= ` 0 )
29		eluzfz1 11841	. . . . . . . . . . 11 |- ( 3 e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... 3 ) )
30	28, 29	ax-mp 5	. . . . . . . . . 10 |- 0 e. ( 0 ... 3 )
31	30, 21	eleqtrri 2539	. . . . . . . . 9 |- 0 e. V
32		2nn 10801	. . . . . . . . . 10 |- 2 e. NN
33		1nn 10653	. . . . . . . . . 10 |- 1 e. NN
34		2t1e2 10792	. . . . . . . . . . . 12 |- ( 2 x. 1 ) = 2
35	34	oveq1i 6330	. . . . . . . . . . 11 |- ( ( 2 x. 1 ) + 1 ) = ( 2 + 1 )
36	35, 11	eqtr4i 2487	. . . . . . . . . 10 |- ( ( 2 x. 1 ) + 1 ) = 3
37		1lt2 10810	. . . . . . . . . 10 |- 1 < 2
38	32, 14, 33, 36, 37	ndvdsi 14446	. . . . . . . . 9 |- -. 2 || 3
39		fveq2 5892	. . . . . . . . . . . . 13 |- ( x = 0 -> ( ( V VDeg E ) ` x ) = ( ( V VDeg E ) ` 0 ) )
40	23	elexi 3067	. . . . . . . . . . . . . 14 |- V e. _V
41		df-s6 12991	. . . . . . . . . . . . . . 15 |- <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> = ( <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } "> ++ <" { 2 , 3 } "> )
42		df-s5 12990	. . . . . . . . . . . . . . . 16 |- <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } "> = ( <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> ++ <" { 1 , 2 } "> )
43		df-s4 12989	. . . . . . . . . . . . . . . . 17 |- <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> = ( <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> ++ <" { 1 , 2 } "> )
44		df-s3 12988	. . . . . . . . . . . . . . . . . 18 |- <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> = ( <" { 0 , 1 } { 0 , 2 } "> ++ <" { 0 , 3 } "> )
45		df-s2 12987	. . . . . . . . . . . . . . . . . . 19 |- <" { 0 , 1 } { 0 , 2 } "> = ( <" { 0 , 1 } "> ++ <" { 0 , 2 } "> )
46		1le3 10860	. . . . . . . . . . . . . . . . . . . . . . 23 |- 1 <_ 3
47	14, 27	eleqtri 2538	. . . . . . . . . . . . . . . . . . . . . . . 24 |- 1 e. ( ZZ>= ` 0 )
48		3z 11004	. . . . . . . . . . . . . . . . . . . . . . . 24 |- 3 e. ZZ
49		elfz5 11827	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( 1 e. ( ZZ>= ` 0 ) /\ 3 e. ZZ ) -> ( 1 e. ( 0 ... 3 ) <-> 1 <_ 3 ) )
50	47, 48, 49	mp2an 683	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( 1 e. ( 0 ... 3 ) <-> 1 <_ 3 )
51	46, 50	mpbir 214	. . . . . . . . . . . . . . . . . . . . . 22 |- 1 e. ( 0 ... 3 )
52	51, 21	eleqtrri 2539	. . . . . . . . . . . . . . . . . . . . 21 |- 1 e. V
53	40, 31, 52	umgrabi 25767	. . . . . . . . . . . . . . . . . . . 20 |- ( T. -> { 0 , 1 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
54	53	s1cld 12784	. . . . . . . . . . . . . . . . . . 19 |- ( T. -> <" { 0 , 1 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
55		2re 10712	. . . . . . . . . . . . . . . . . . . . . . 23 |- 2 e. RR
56		3re 10716	. . . . . . . . . . . . . . . . . . . . . . 23 |- 3 e. RR
57		2lt3 10811	. . . . . . . . . . . . . . . . . . . . . . 23 |- 2 < 3
58	55, 56, 57	ltleii 9788	. . . . . . . . . . . . . . . . . . . . . 22 |- 2 <_ 3
59		2eluzge0 11237	. . . . . . . . . . . . . . . . . . . . . . 23 |- 2 e. ( ZZ>= ` 0 )
60		elfz5 11827	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( 2 e. ( ZZ>= ` 0 ) /\ 3 e. ZZ ) -> ( 2 e. ( 0 ... 3 ) <-> 2 <_ 3 ) )
61	59, 48, 60	mp2an 683	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 2 e. ( 0 ... 3 ) <-> 2 <_ 3 )
62	58, 61	mpbir 214	. . . . . . . . . . . . . . . . . . . . 21 |- 2 e. ( 0 ... 3 )
63	62, 21	eleqtrri 2539	. . . . . . . . . . . . . . . . . . . 20 |- 2 e. V
64	40, 31, 63	umgrabi 25767	. . . . . . . . . . . . . . . . . . 19 |- ( T. -> { 0 , 2 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
65	45, 54, 64	cats1cld 12994	. . . . . . . . . . . . . . . . . 18 |- ( T. -> <" { 0 , 1 } { 0 , 2 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
66		eluzfz2 11842	. . . . . . . . . . . . . . . . . . . . 21 |- ( 3 e. ( ZZ>= ` 0 ) -> 3 e. ( 0 ... 3 ) )
67	28, 66	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 |- 3 e. ( 0 ... 3 )
68	67, 21	eleqtrri 2539	. . . . . . . . . . . . . . . . . . 19 |- 3 e. V
69	40, 31, 68	umgrabi 25767	. . . . . . . . . . . . . . . . . 18 |- ( T. -> { 0 , 3 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
70	44, 65, 69	cats1cld 12994	. . . . . . . . . . . . . . . . 17 |- ( T. -> <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
71	40, 52, 63	umgrabi 25767	. . . . . . . . . . . . . . . . 17 |- ( T. -> { 1 , 2 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
72	43, 70, 71	cats1cld 12994	. . . . . . . . . . . . . . . 16 |- ( T. -> <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
73	42, 72, 71	cats1cld 12994	. . . . . . . . . . . . . . 15 |- ( T. -> <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
74	40, 63, 68	umgrabi 25767	. . . . . . . . . . . . . . 15 |- ( T. -> { 2 , 3 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
75	41, 73, 74	cats1cld 12994	. . . . . . . . . . . . . 14 |- ( T. -> <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
76		wrd0 12733	. . . . . . . . . . . . . . . . . . . . 21 |- (/) e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 }
77	76	a1i 11	. . . . . . . . . . . . . . . . . . . 20 |- ( T. -> (/) e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
78	40, 31	vdeg0i 25766	. . . . . . . . . . . . . . . . . . . 20 |- ( ( V VDeg (/) ) ` 0 ) = 0
79		1e0p1 11113	. . . . . . . . . . . . . . . . . . . 20 |- 1 = ( 0 + 1 )
80		ax-1ne0 9639	. . . . . . . . . . . . . . . . . . . 20 |- 1 =/= 0
81		s0s1 13059	. . . . . . . . . . . . . . . . . . . 20 |- <" { 0 , 1 } "> = ( (/) ++ <" { 0 , 1 } "> )
82	40, 77, 31, 78, 79, 52, 80, 81	vdegp1bi 25769	. . . . . . . . . . . . . . . . . . 19 |- ( ( V VDeg <" { 0 , 1 } "> ) ` 0 ) = 1
83		df-2 10701	. . . . . . . . . . . . . . . . . . 19 |- 2 = ( 1 + 1 )
84		2ne0 10735	. . . . . . . . . . . . . . . . . . 19 |- 2 =/= 0
85	40, 54, 31, 82, 83, 63, 84, 45	vdegp1bi 25769	. . . . . . . . . . . . . . . . . 18 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } "> ) ` 0 ) = 2
86	40, 65, 31, 85, 11, 68, 2, 44	vdegp1bi 25769	. . . . . . . . . . . . . . . . 17 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> ) ` 0 ) = 3
87	40, 70, 31, 86, 52, 80, 63, 84, 43	vdegp1ai 25768	. . . . . . . . . . . . . . . 16 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> ) ` 0 ) = 3
88	40, 72, 31, 87, 52, 80, 63, 84, 42	vdegp1ai 25768	. . . . . . . . . . . . . . 15 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } "> ) ` 0 ) = 3
89	40, 73, 31, 88, 63, 84, 68, 2, 41	vdegp1ai 25768	. . . . . . . . . . . . . 14 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> ) ` 0 ) = 3
90		konigsberg.e	. . . . . . . . . . . . . . 15 |- E = <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } { 2 , 3 } ">
91		df-s7 12992	. . . . . . . . . . . . . . 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 } "> ++ <" { 2 , 3 } "> )
92	90, 91	eqtri 2484	. . . . . . . . . . . . . 14 |- E = ( <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> ++ <" { 2 , 3 } "> )
93	40, 75, 31, 89, 63, 84, 68, 2, 92	vdegp1ai 25768	. . . . . . . . . . . . 13 |- ( ( V VDeg E ) ` 0 ) = 3
94	39, 93	syl6eq 2512	. . . . . . . . . . . 12 |- ( x = 0 -> ( ( V VDeg E ) ` x ) = 3 )
95	94	breq2d 4430	. . . . . . . . . . 11 |- ( x = 0 -> ( 2 || ( ( V VDeg E ) ` x ) <-> 2 || 3 ) )
96	95	notbid 300	. . . . . . . . . 10 |- ( x = 0 -> ( -. 2 || ( ( V VDeg E ) ` x ) <-> -. 2 || 3 ) )
97	96	elrab 3208	. . . . . . . . 9 |- ( 0 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } <-> ( 0 e. V /\ -. 2 || 3 ) )
98	31, 38, 97	mpbir2an 936	. . . . . . . 8 |- 0 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
99		fveq2 5892	. . . . . . . . . . . . 13 |- ( x = 1 -> ( ( V VDeg E ) ` x ) = ( ( V VDeg E ) ` 1 ) )
100	40, 52	vdeg0i 25766	. . . . . . . . . . . . . . . . . . . 20 |- ( ( V VDeg (/) ) ` 1 ) = 0
101	40, 77, 52, 100, 79, 31, 12, 81	vdegp1ci 25770	. . . . . . . . . . . . . . . . . . 19 |- ( ( V VDeg <" { 0 , 1 } "> ) ` 1 ) = 1
102	3, 37	gtneii 9777	. . . . . . . . . . . . . . . . . . 19 |- 2 =/= 1
103	40, 54, 52, 101, 31, 12, 63, 102, 45	vdegp1ai 25768	. . . . . . . . . . . . . . . . . 18 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } "> ) ` 1 ) = 1
104	40, 65, 52, 103, 31, 12, 68, 5, 44	vdegp1ai 25768	. . . . . . . . . . . . . . . . 17 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> ) ` 1 ) = 1
105	40, 70, 52, 104, 83, 63, 102, 43	vdegp1bi 25769	. . . . . . . . . . . . . . . 16 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> ) ` 1 ) = 2
106	40, 72, 52, 105, 11, 63, 102, 42	vdegp1bi 25769	. . . . . . . . . . . . . . 15 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } "> ) ` 1 ) = 3
107	40, 73, 52, 106, 63, 102, 68, 5, 41	vdegp1ai 25768	. . . . . . . . . . . . . 14 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> ) ` 1 ) = 3
108	40, 75, 52, 107, 63, 102, 68, 5, 92	vdegp1ai 25768	. . . . . . . . . . . . 13 |- ( ( V VDeg E ) ` 1 ) = 3
109	99, 108	syl6eq 2512	. . . . . . . . . . . 12 |- ( x = 1 -> ( ( V VDeg E ) ` x ) = 3 )
110	109	breq2d 4430	. . . . . . . . . . 11 |- ( x = 1 -> ( 2 || ( ( V VDeg E ) ` x ) <-> 2 || 3 ) )
111	110	notbid 300	. . . . . . . . . 10 |- ( x = 1 -> ( -. 2 || ( ( V VDeg E ) ` x ) <-> -. 2 || 3 ) )
112	111	elrab 3208	. . . . . . . . 9 |- ( 1 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } <-> ( 1 e. V /\ -. 2 || 3 ) )
113	52, 38, 112	mpbir2an 936	. . . . . . . 8 |- 1 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
114		prssi 4141	. . . . . . . 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 ) } )
115	98, 113, 114	mp2an 683	. . . . . . 7 |- { 0 , 1 } C_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
116		fveq2 5892	. . . . . . . . . . . . 13 |- ( x = 3 -> ( ( V VDeg E ) ` x ) = ( ( V VDeg E ) ` 3 ) )
117	40, 68	vdeg0i 25766	. . . . . . . . . . . . . . . . . . . 20 |- ( ( V VDeg (/) ) ` 3 ) = 0
118		0re 9674	. . . . . . . . . . . . . . . . . . . . 21 |- 0 e. RR
119		3pos 10736	. . . . . . . . . . . . . . . . . . . . 21 |- 0 < 3
120	118, 119	ltneii 9778	. . . . . . . . . . . . . . . . . . . 20 |- 0 =/= 3
121	3, 4	ltneii 9778	. . . . . . . . . . . . . . . . . . . 20 |- 1 =/= 3
122	40, 77, 68, 117, 31, 120, 52, 121, 81	vdegp1ai 25768	. . . . . . . . . . . . . . . . . . 19 |- ( ( V VDeg <" { 0 , 1 } "> ) ` 3 ) = 0
123	55, 57	ltneii 9778	. . . . . . . . . . . . . . . . . . 19 |- 2 =/= 3
124	40, 54, 68, 122, 31, 120, 63, 123, 45	vdegp1ai 25768	. . . . . . . . . . . . . . . . . 18 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } "> ) ` 3 ) = 0
125	40, 65, 68, 124, 79, 31, 120, 44	vdegp1ci 25770	. . . . . . . . . . . . . . . . 17 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> ) ` 3 ) = 1
126	40, 70, 68, 125, 52, 121, 63, 123, 43	vdegp1ai 25768	. . . . . . . . . . . . . . . 16 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> ) ` 3 ) = 1
127	40, 72, 68, 126, 52, 121, 63, 123, 42	vdegp1ai 25768	. . . . . . . . . . . . . . 15 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } "> ) ` 3 ) = 1
128	40, 73, 68, 127, 83, 63, 123, 41	vdegp1ci 25770	. . . . . . . . . . . . . 14 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> ) ` 3 ) = 2
129	40, 75, 68, 128, 11, 63, 123, 92	vdegp1ci 25770	. . . . . . . . . . . . 13 |- ( ( V VDeg E ) ` 3 ) = 3
130	116, 129	syl6eq 2512	. . . . . . . . . . . 12 |- ( x = 3 -> ( ( V VDeg E ) ` x ) = 3 )
131	130	breq2d 4430	. . . . . . . . . . 11 |- ( x = 3 -> ( 2 || ( ( V VDeg E ) ` x ) <-> 2 || 3 ) )
132	131	notbid 300	. . . . . . . . . 10 |- ( x = 3 -> ( -. 2 || ( ( V VDeg E ) ` x ) <-> -. 2 || 3 ) )
133	132	elrab 3208	. . . . . . . . 9 |- ( 3 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } <-> ( 3 e. V /\ -. 2 || 3 ) )
134	68, 38, 133	mpbir2an 936	. . . . . . . 8 |- 3 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
135		snssi 4129	. . . . . . . 8 |- ( 3 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } -> { 3 } C_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } )
136	134, 135	ax-mp 5	. . . . . . 7 |- { 3 } C_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
137	115, 136	unssi 3621	. . . . . 6 |- ( { 0 , 1 } u. { 3 } ) C_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
138		ssdomg 7646	. . . . . 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 ) } ) )
139	26, 137, 138	mp2 9	. . . . 5 |- ( { 0 , 1 } u. { 3 } ) ~<_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
140		snfi 7681	. . . . . . 7 |- { 3 } e. Fin
141		unfi 7869	. . . . . . 7 |- ( ( { 0 , 1 } e. Fin /\ { 3 } e. Fin ) -> ( { 0 , 1 } u. { 3 } ) e. Fin )
142	1, 140, 141	mp2an 683	. . . . . 6 |- ( { 0 , 1 } u. { 3 } ) e. Fin
143		hashdom 12596	. . . . . 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 ) } ) )
144	142, 26, 143	mp2an 683	. . . . 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 ) } )
145	139, 144	mpbir 214	. . . 4 |- ( # ` ( { 0 , 1 } u. { 3 } ) ) <_ ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } )
146	20, 145	eqbrtrri 4440	. . 3 |- 3 <_ ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } )
147		hashcl 12576	. . . . . 6 |- ( { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } e. Fin -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) e. NN0 )
148	26, 147	ax-mp 5	. . . . 5 |- ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) e. NN0
149	148	nn0rei 10914	. . . 4 |- ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) e. RR
150	56, 149	lenlti 9785	. . 3 |- ( 3 <_ ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) <-> -. ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) < 3 )
151	146, 150	mpbi 213	. 2 |- -. ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) < 3
152		eupath 25765	. . . 4 |- ( ( V EulPaths E ) =/= (/) -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) e. { 0 , 2 } )
153		elpri 3997	. . . 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 ) )
154		id 22	. . . . . 6 |- ( ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 0 -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 0 )
155	154, 119	syl6eqbr 4456	. . . . 5 |- ( ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 0 -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) < 3 )
156		id 22	. . . . . 6 |- ( ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 2 -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 2 )
157	156, 57	syl6eqbr 4456	. . . . 5 |- ( ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) = 2 -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) < 3 )
158	155, 157	jaoi 385	. . . 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 )
159	152, 153, 158	3syl 18	. . 3 |- ( ( V EulPaths E ) =/= (/) -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) < 3 )
160	159	necon1bi 2664	. 2 |- ( -. ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) < 3 -> ( V EulPaths E ) = (/) )
161	151, 160	ax-mp 5	1 |- ( V EulPaths E ) = (/)

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   = wceq 1455   T. wtru 1456   e. wcel 1898   =/= wne 2633   {crab 2753   \ cdif 3413   u. cun 3414   C_ wss 3416   (/)c0 3743   ~Pcpw 3963   {csn 3980   {cpr 3982   class class class wbr 4418   ` cfv 5605  (class class class)co 6320   ~<_ cdom 7598   Fincfn 7600   0cc0 9570   1c1 9571   + caddc 9573   x. cmul 9575   < clt 9706   <_ cle 9707   2c2 10692   3c3 10693   NN0cn0 10903   ZZcz 10971   ZZ>=cuz 11193   ...cfz 11819   #chash 12553  Word cword 12695   ++ cconcat 12697   <"cs1 12698   <"cs2 12980   <"cs3 12981   <"cs4 12982   <"cs5 12983   <"cs6 12984   <"cs7 12985   || cdvds 14360   VDeg cvdg 25677   EulPaths ceup 25746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647  ax-pre-sup 9648

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-1o 7213  df-oadd 7217  df-er 7394  df-pm 7506  df-en 7601  df-dom 7602  df-sdom 7603  df-fin 7604  df-sup 7987  df-inf 7988  df-card 8404  df-cda 8629  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-div 10303  df-nn 10643  df-2 10701  df-3 10702  df-n0 10904  df-z 10972  df-uz 11194  df-rp 11337  df-xadd 11444  df-fz 11820  df-fzo 11953  df-seq 12252  df-exp 12311  df-hash 12554  df-word 12703  df-concat 12705  df-s1 12706  df-s2 12987  df-s3 12988  df-s4 12989  df-s5 12990  df-s6 12991  df-s7 12992  df-cj 13217  df-re 13218  df-im 13219  df-sqrt 13353  df-abs 13354  df-dvds 14361  df-umgra 25096  df-vdgr 25678  df-eupa 25747

This theorem is referenced by: (None)