Theorem konigsberg 24679

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 24673 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 7794	. . . . . 6 |- { 0 , 1 } e. Fin
2		3ne0 10629	. . . . . . 7 |- 3 =/= 0
3		1re 9594	. . . . . . . 8 |- 1 e. RR
4		1lt3 10703	. . . . . . . 8 |- 1 < 3
5	3, 4	gtneii 9695	. . . . . . 7 |- 3 =/= 1
6	2, 5	nelpri 4048	. . . . . 6 |- -. 3 e. { 0 , 1 }
7		3nn0 10812	. . . . . . 7 |- 3 e. NN0
8		hashunsng 12426	. . . . . . 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 10594	. . . . . 6 |- 3 = ( 2 + 1 )
12		0ne1 10602	. . . . . . . 8 |- 0 =/= 1
13		0nn0 10809	. . . . . . . . 9 |- 0 e. NN0
14		1nn0 10810	. . . . . . . . 9 |- 1 e. NN0
15		hashprg 12427	. . . . . . . . 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 6293	. . . . . 6 |- ( ( # ` { 0 , 1 } ) + 1 ) = ( 2 + 1 )
19	11, 18	eqtr4i 2499	. . . . 5 |- 3 = ( ( # ` { 0 , 1 } ) + 1 )
20	10, 19	eqtr4i 2499	. . . 4 |- ( # ` ( { 0 , 1 } u. { 3 } ) ) = 3
21		konigsberg.v	. . . . . . . 8 |- V = ( 0 ... 3 )
22		fzfi 12049	. . . . . . . 8 |- ( 0 ... 3 ) e. Fin
23	21, 22	eqeltri 2551	. . . . . . 7 |- V e. Fin
24		ssrab2 3585	. . . . . . 7 |- { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } C_ V
25		ssfi 7740	. . . . . . 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 11115	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
28	7, 27	eleqtri 2553	. . . . . . . . . . 11 |- 3 e. ( ZZ>= ` 0 )
29		eluzfz1 11692	. . . . . . . . . . 11 |- ( 3 e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... 3 ) )
30	28, 29	ax-mp 5	. . . . . . . . . 10 |- 0 e. ( 0 ... 3 )
31	30, 21	eleqtrri 2554	. . . . . . . . 9 |- 0 e. V
32		2nn 10692	. . . . . . . . . 10 |- 2 e. NN
33		1nn 10546	. . . . . . . . . 10 |- 1 e. NN
34		2t1e2 10683	. . . . . . . . . . . 12 |- ( 2 x. 1 ) = 2
35	34	oveq1i 6293	. . . . . . . . . . 11 |- ( ( 2 x. 1 ) + 1 ) = ( 2 + 1 )
36	35, 11	eqtr4i 2499	. . . . . . . . . 10 |- ( ( 2 x. 1 ) + 1 ) = 3
37		1lt2 10701	. . . . . . . . . 10 |- 1 < 2
38	32, 14, 33, 36, 37	ndvdsi 13926	. . . . . . . . 9 |- -. 2 || 3
39		fveq2 5865	. . . . . . . . . . . . 13 |- ( x = 0 -> ( ( V VDeg E ) ` x ) = ( ( V VDeg E ) ` 0 ) )
40	23	elexi 3123	. . . . . . . . . . . . . 14 |- V e. _V
41		df-s6 12779	. . . . . . . . . . . . . . 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 12778	. . . . . . . . . . . . . . . 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 12777	. . . . . . . . . . . . . . . . 17 |- <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> = ( <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> concat <" { 1 , 2 } "> )
44		df-s3 12776	. . . . . . . . . . . . . . . . . 18 |- <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> = ( <" { 0 , 1 } { 0 , 2 } "> concat <" { 0 , 3 } "> )
45		df-s2 12775	. . . . . . . . . . . . . . . . . . 19 |- <" { 0 , 1 } { 0 , 2 } "> = ( <" { 0 , 1 } "> concat <" { 0 , 2 } "> )
46		1le3 10751	. . . . . . . . . . . . . . . . . . . . . . 23 |- 1 <_ 3
47	14, 27	eleqtri 2553	. . . . . . . . . . . . . . . . . . . . . . . 24 |- 1 e. ( ZZ>= ` 0 )
48		3z 10896	. . . . . . . . . . . . . . . . . . . . . . . 24 |- 3 e. ZZ
49		elfz5 11679	. . . . . . . . . . . . . . . . . . . . . . . 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 2554	. . . . . . . . . . . . . . . . . . . . 21 |- 1 e. V
53	40, 31, 52	umgrabi 24675	. . . . . . . . . . . . . . . . . . . 20 |- ( T. -> { 0 , 1 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
54	53	s1cld 12577	. . . . . . . . . . . . . . . . . . 19 |- ( T. -> <" { 0 , 1 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
55		2re 10604	. . . . . . . . . . . . . . . . . . . . . . 23 |- 2 e. RR
56		3re 10608	. . . . . . . . . . . . . . . . . . . . . . 23 |- 3 e. RR
57		2lt3 10702	. . . . . . . . . . . . . . . . . . . . . . 23 |- 2 < 3
58	55, 56, 57	ltleii 9706	. . . . . . . . . . . . . . . . . . . . . 22 |- 2 <_ 3
59		2nn0 10811	. . . . . . . . . . . . . . . . . . . . . . . 24 |- 2 e. NN0
60	59, 27	eleqtri 2553	. . . . . . . . . . . . . . . . . . . . . . 23 |- 2 e. ( ZZ>= ` 0 )
61		elfz5 11679	. . . . . . . . . . . . . . . . . . . . . . 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 2554	. . . . . . . . . . . . . . . . . . . 20 |- 2 e. V
65	40, 31, 64	umgrabi 24675	. . . . . . . . . . . . . . . . . . 19 |- ( T. -> { 0 , 2 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
66	45, 54, 65	cats1cld 12782	. . . . . . . . . . . . . . . . . 18 |- ( T. -> <" { 0 , 1 } { 0 , 2 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
67		eluzfz2 11693	. . . . . . . . . . . . . . . . . . . . 21 |- ( 3 e. ( ZZ>= ` 0 ) -> 3 e. ( 0 ... 3 ) )
68	28, 67	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 |- 3 e. ( 0 ... 3 )
69	68, 21	eleqtrri 2554	. . . . . . . . . . . . . . . . . . 19 |- 3 e. V
70	40, 31, 69	umgrabi 24675	. . . . . . . . . . . . . . . . . 18 |- ( T. -> { 0 , 3 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
71	44, 66, 70	cats1cld 12782	. . . . . . . . . . . . . . . . 17 |- ( T. -> <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
72	40, 52, 64	umgrabi 24675	. . . . . . . . . . . . . . . . 17 |- ( T. -> { 1 , 2 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
73	43, 71, 72	cats1cld 12782	. . . . . . . . . . . . . . . 16 |- ( T. -> <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
74	42, 73, 72	cats1cld 12782	. . . . . . . . . . . . . . 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 24675	. . . . . . . . . . . . . . 15 |- ( T. -> { 2 , 3 } e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
76	41, 74, 75	cats1cld 12782	. . . . . . . . . . . . . 14 |- ( T. -> <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } { 1 , 2 } { 2 , 3 } "> e. Word { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
77		wrd0 12530	. . . . . . . . . . . . . . . . . . . . 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 24674	. . . . . . . . . . . . . . . . . . . 20 |- ( ( V VDeg (/) ) ` 0 ) = 0
80		1e0p1 11003	. . . . . . . . . . . . . . . . . . . 20 |- 1 = ( 0 + 1 )
81		ax-1ne0 9560	. . . . . . . . . . . . . . . . . . . 20 |- 1 =/= 0
82		s0s1 12832	. . . . . . . . . . . . . . . . . . . 20 |- <" { 0 , 1 } "> = ( (/) concat <" { 0 , 1 } "> )
83	40, 78, 31, 79, 80, 52, 81, 82	vdegp1bi 24677	. . . . . . . . . . . . . . . . . . 19 |- ( ( V VDeg <" { 0 , 1 } "> ) ` 0 ) = 1
84		df-2 10593	. . . . . . . . . . . . . . . . . . 19 |- 2 = ( 1 + 1 )
85		2ne0 10627	. . . . . . . . . . . . . . . . . . 19 |- 2 =/= 0
86	40, 54, 31, 83, 84, 64, 85, 45	vdegp1bi 24677	. . . . . . . . . . . . . . . . . 18 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } "> ) ` 0 ) = 2
87	40, 66, 31, 86, 11, 69, 2, 44	vdegp1bi 24677	. . . . . . . . . . . . . . . . 17 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> ) ` 0 ) = 3
88	40, 71, 31, 87, 52, 81, 64, 85, 43	vdegp1ai 24676	. . . . . . . . . . . . . . . 16 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> ) ` 0 ) = 3
89	40, 73, 31, 88, 52, 81, 64, 85, 42	vdegp1ai 24676	. . . . . . . . . . . . . . 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 24676	. . . . . . . . . . . . . 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 12780	. . . . . . . . . . . . . . 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 2496	. . . . . . . . . . . . . 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 24676	. . . . . . . . . . . . 13 |- ( ( V VDeg E ) ` 0 ) = 3
95	39, 94	syl6eq 2524	. . . . . . . . . . . 12 |- ( x = 0 -> ( ( V VDeg E ) ` x ) = 3 )
96	95	breq2d 4459	. . . . . . . . . . 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 3261	. . . . . . . . 9 |- ( 0 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } <-> ( 0 e. V /\ -. 2 || 3 ) )
99	31, 38, 98	mpbir2an 918	. . . . . . . 8 |- 0 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
100		fveq2 5865	. . . . . . . . . . . . 13 |- ( x = 1 -> ( ( V VDeg E ) ` x ) = ( ( V VDeg E ) ` 1 ) )
101	40, 52	vdeg0i 24674	. . . . . . . . . . . . . . . . . . . 20 |- ( ( V VDeg (/) ) ` 1 ) = 0
102	40, 78, 52, 101, 80, 31, 12, 82	vdegp1ci 24678	. . . . . . . . . . . . . . . . . . 19 |- ( ( V VDeg <" { 0 , 1 } "> ) ` 1 ) = 1
103	3, 37	gtneii 9695	. . . . . . . . . . . . . . . . . . 19 |- 2 =/= 1
104	40, 54, 52, 102, 31, 12, 64, 103, 45	vdegp1ai 24676	. . . . . . . . . . . . . . . . . 18 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } "> ) ` 1 ) = 1
105	40, 66, 52, 104, 31, 12, 69, 5, 44	vdegp1ai 24676	. . . . . . . . . . . . . . . . 17 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> ) ` 1 ) = 1
106	40, 71, 52, 105, 84, 64, 103, 43	vdegp1bi 24677	. . . . . . . . . . . . . . . 16 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> ) ` 1 ) = 2
107	40, 73, 52, 106, 11, 64, 103, 42	vdegp1bi 24677	. . . . . . . . . . . . . . 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 24676	. . . . . . . . . . . . . 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 24676	. . . . . . . . . . . . 13 |- ( ( V VDeg E ) ` 1 ) = 3
110	100, 109	syl6eq 2524	. . . . . . . . . . . 12 |- ( x = 1 -> ( ( V VDeg E ) ` x ) = 3 )
111	110	breq2d 4459	. . . . . . . . . . 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 3261	. . . . . . . . 9 |- ( 1 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } <-> ( 1 e. V /\ -. 2 || 3 ) )
114	52, 38, 113	mpbir2an 918	. . . . . . . 8 |- 1 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
115		prssi 4183	. . . . . . . 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 5865	. . . . . . . . . . . . 13 |- ( x = 3 -> ( ( V VDeg E ) ` x ) = ( ( V VDeg E ) ` 3 ) )
118	40, 69	vdeg0i 24674	. . . . . . . . . . . . . . . . . . . 20 |- ( ( V VDeg (/) ) ` 3 ) = 0
119		0re 9595	. . . . . . . . . . . . . . . . . . . . 21 |- 0 e. RR
120		3pos 10628	. . . . . . . . . . . . . . . . . . . . 21 |- 0 < 3
121	119, 120	ltneii 9696	. . . . . . . . . . . . . . . . . . . 20 |- 0 =/= 3
122	3, 4	ltneii 9696	. . . . . . . . . . . . . . . . . . . 20 |- 1 =/= 3
123	40, 78, 69, 118, 31, 121, 52, 122, 82	vdegp1ai 24676	. . . . . . . . . . . . . . . . . . 19 |- ( ( V VDeg <" { 0 , 1 } "> ) ` 3 ) = 0
124	55, 57	ltneii 9696	. . . . . . . . . . . . . . . . . . 19 |- 2 =/= 3
125	40, 54, 69, 123, 31, 121, 64, 124, 45	vdegp1ai 24676	. . . . . . . . . . . . . . . . . 18 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } "> ) ` 3 ) = 0
126	40, 66, 69, 125, 80, 31, 121, 44	vdegp1ci 24678	. . . . . . . . . . . . . . . . 17 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } "> ) ` 3 ) = 1
127	40, 71, 69, 126, 52, 122, 64, 124, 43	vdegp1ai 24676	. . . . . . . . . . . . . . . 16 |- ( ( V VDeg <" { 0 , 1 } { 0 , 2 } { 0 , 3 } { 1 , 2 } "> ) ` 3 ) = 1
128	40, 73, 69, 127, 52, 122, 64, 124, 42	vdegp1ai 24676	. . . . . . . . . . . . . . 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 24678	. . . . . . . . . . . . . 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 24678	. . . . . . . . . . . . 13 |- ( ( V VDeg E ) ` 3 ) = 3
131	117, 130	syl6eq 2524	. . . . . . . . . . . 12 |- ( x = 3 -> ( ( V VDeg E ) ` x ) = 3 )
132	131	breq2d 4459	. . . . . . . . . . 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 3261	. . . . . . . . 9 |- ( 3 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } <-> ( 3 e. V /\ -. 2 || 3 ) )
135	69, 38, 134	mpbir2an 918	. . . . . . . 8 |- 3 e. { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
136		snssi 4171	. . . . . . . 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 3679	. . . . . 6 |- ( { 0 , 1 } u. { 3 } ) C_ { x e. V | -. 2 || ( ( V VDeg E ) ` x ) }
139		ssdomg 7561	. . . . . 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 7596	. . . . . . 7 |- { 3 } e. Fin
142		unfi 7786	. . . . . . 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 12414	. . . . . 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 4468	. . 3 |- 3 <_ ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } )
148		hashcl 12395	. . . . . 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 10805	. . . 4 |- ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) e. RR
151	56, 150	lenlti 9703	. . 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 24673	. . . 4 |- ( ( V EulPaths E ) =/= (/) -> ( # ` { x e. V | -. 2 || ( ( V VDeg E ) ` x ) } ) e. { 0 , 2 } )
154		elpri 4047	. . . 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 4484	. . . . 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 4484	. . . . 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 2700	. 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 1379   T. wtru 1380   e. wcel 1767   =/= wne 2662   {crab 2818   \ cdif 3473   u. cun 3474   C_ wss 3476   (/)c0 3785   ~Pcpw 4010   {csn 4027   {cpr 4029   class class class wbr 4447   ` cfv 5587  (class class class)co 6283   ~<_ cdom 7514   Fincfn 7516   0cc0 9491   1c1 9492   + caddc 9494   x. cmul 9496   < clt 9627   <_ cle 9628   2c2 10584   3c3 10585   NN0cn0 10794   ZZcz 10863   ZZ>=cuz 11081   ...cfz 11671   #chash 12372  Word cword 12499   concat cconcat 12501   <"cs1 12502   <"cs2 12768   <"cs3 12769   <"cs4 12770   <"cs5 12771   <"cs6 12772   <"cs7 12773   || cdivides 13846   VDeg cvdg 24585   EulPaths ceup 24654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-card 8319  df-cda 8547  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-n0 10795  df-z 10864  df-uz 11082  df-rp 11220  df-xadd 11318  df-fz 11672  df-fzo 11792  df-seq 12075  df-exp 12134  df-hash 12373  df-word 12507  df-concat 12509  df-s1 12510  df-s2 12775  df-s3 12776  df-s4 12777  df-s5 12778  df-s6 12779  df-s7 12780  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-dvds 13847  df-umgra 24005  df-vdgr 24586  df-eupa 24655

This theorem is referenced by: (None)