Theorem clwwlk0on0 16443

Description: There is no word over the set of vertices representing a closed walk on vertex X of length 0 in a graph G. (Contributed by AV, 17-Feb-2022.) (Revised by AV, 25-Feb-2022.)

Assertion

Ref	Expression
clwwlk0on0	|- ( X (ClWWalksNOn ` G ) 0 ) = (/)

Proof of Theorem clwwlk0on0

Dummy variables n v w x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwwlknonmpo 16440	. . . 4 |- (ClWWalksNOn ` G ) = ( v e. (Vtx ` G ) , n e. NN0 |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
2	1	elmpocl1 6252	. . 3 |- ( x e. ( X (ClWWalksNOn ` G ) 0 ) -> X e. (Vtx ` G ) )
3		noel 3514	. . . 4 |- -. x e. (/)
4	3	pm2.21i 651	. . 3 |- ( x e. (/) -> X e. (Vtx ` G ) )
5		0nn0 9513	. . . . 5 |- 0 e. NN0
6		eqeq2 2244	. . . . . . . 8 |- ( v = X -> ( ( w ` 0 ) = v <-> ( w ` 0 ) = X ) )
7	6	rabbidv 2804	. . . . . . 7 |- ( v = X -> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } = { w e. ( n ClWWalksN G ) | ( w ` 0 ) = X } )
8		oveq1 6059	. . . . . . . . 9 |- ( n = 0 -> ( n ClWWalksN G ) = ( 0 ClWWalksN G ) )
9		clwwlkn0 16420	. . . . . . . . 9 |- ( 0 ClWWalksN G ) = (/)
10	8, 9	eqtrdi 2283	. . . . . . . 8 |- ( n = 0 -> ( n ClWWalksN G ) = (/) )
11	10	rabeqdv 2809	. . . . . . 7 |- ( n = 0 -> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = X } = { w e. (/) | ( w ` 0 ) = X } )
12		0ex 4239	. . . . . . . 8 |- (/) e. _V
13	12	rabex 4258	. . . . . . 7 |- { w e. (/) | ( w ` 0 ) = X } e. _V
14	7, 11, 1, 13	ovmpo 6191	. . . . . 6 |- ( ( X e. (Vtx ` G ) /\ 0 e. NN0 ) -> ( X (ClWWalksNOn ` G ) 0 ) = { w e. (/) | ( w ` 0 ) = X } )
15		rab0 3539	. . . . . 6 |- { w e. (/) | ( w ` 0 ) = X } = (/)
16	14, 15	eqtrdi 2283	. . . . 5 |- ( ( X e. (Vtx ` G ) /\ 0 e. NN0 ) -> ( X (ClWWalksNOn ` G ) 0 ) = (/) )
17	5, 16	mpan2 425	. . . 4 |- ( X e. (Vtx ` G ) -> ( X (ClWWalksNOn ` G ) 0 ) = (/) )
18	17	eleq2d 2304	. . 3 |- ( X e. (Vtx ` G ) -> ( x e. ( X (ClWWalksNOn ` G ) 0 ) <-> x e. (/) ) )
19	2, 4, 18	pm5.21nii 712	. 2 |- ( x e. ( X (ClWWalksNOn ` G ) 0 ) <-> x e. (/) )
20	19	eqriv 2231	1 |- ( X (ClWWalksNOn ` G ) 0 ) = (/)

Syntax hints:   /\ wa 104   = wceq 1398   e. wcel 2205   {crab 2526   (/)c0 3510   ` cfv 5354  (class class class)co 6052   0cc0 8129   NN0cn0 9498  Vtxcvtx 16024   ClWWalksN cclwwlkn 16415  ClWWalksNOncclwwlknon 16438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-er 6769  df-map 6886  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-fz 10346  df-fzo 10481  df-ihash 11143  df-word 11229  df-ndx 13232  df-slot 13233  df-base 13235  df-vtx 16026  df-clwwlk 16404  df-clwwlkn 16416  df-clwwlknon 16439

This theorem is referenced by: (None)