Theorem wlkv0 16080

Description: If there is a walk in the null graph (a class without vertices), it would be the pair consisting of empty sets. (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)

Assertion

Ref	Expression
wlkv0	|- ( ( (Vtx ` G ) = (/) /\ W e. (Walks ` G ) ) -> ( ( 1st ` W ) = (/) /\ ( 2nd ` W ) = (/) ) )

Proof of Theorem wlkv0

Step	Hyp	Ref	Expression
1		eqid 2229	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
2	1	wlkf 16042	. . . 4 |- ( ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) -> ( 1st ` W ) e. Word dom (iEdg ` G ) )
3		eqid 2229	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
4	3	wlkp 16046	. . . 4 |- ( ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) -> ( 2nd ` W ) : ( 0 ... (♯ ` ( 1st ` W ) ) ) --> (Vtx ` G ) )
5	2, 4	jca 306	. . 3 |- ( ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) -> ( ( 1st ` W ) e. Word dom (iEdg ` G ) /\ ( 2nd ` W ) : ( 0 ... (♯ ` ( 1st ` W ) ) ) --> (Vtx ` G ) ) )
6		feq3 5458	. . . . . 6 |- ( (Vtx ` G ) = (/) -> ( ( 2nd ` W ) : ( 0 ... (♯ ` ( 1st ` W ) ) ) --> (Vtx ` G ) <-> ( 2nd ` W ) : ( 0 ... (♯ ` ( 1st ` W ) ) ) --> (/) ) )
7		f00 5517	. . . . . 6 |- ( ( 2nd ` W ) : ( 0 ... (♯ ` ( 1st ` W ) ) ) --> (/) <-> ( ( 2nd ` W ) = (/) /\ ( 0 ... (♯ ` ( 1st ` W ) ) ) = (/) ) )
8	6, 7	bitrdi 196	. . . . 5 |- ( (Vtx ` G ) = (/) -> ( ( 2nd ` W ) : ( 0 ... (♯ ` ( 1st ` W ) ) ) --> (Vtx ` G ) <-> ( ( 2nd ` W ) = (/) /\ ( 0 ... (♯ ` ( 1st ` W ) ) ) = (/) ) ) )
9		0z 9457	. . . . . . . . . . . 12 |- 0 e. ZZ
10		nn0z 9466	. . . . . . . . . . . 12 |- ( (♯ ` ( 1st ` W ) ) e. NN0 -> (♯ ` ( 1st ` W ) ) e. ZZ )
11		fzn 10238	. . . . . . . . . . . 12 |- ( ( 0 e. ZZ /\ (♯ ` ( 1st ` W ) ) e. ZZ ) -> ( (♯ ` ( 1st ` W ) ) < 0 <-> ( 0 ... (♯ ` ( 1st ` W ) ) ) = (/) ) )
12	9, 10, 11	sylancr 414	. . . . . . . . . . 11 |- ( (♯ ` ( 1st ` W ) ) e. NN0 -> ( (♯ ` ( 1st ` W ) ) < 0 <-> ( 0 ... (♯ ` ( 1st ` W ) ) ) = (/) ) )
13		nn0nlt0 9395	. . . . . . . . . . . 12 |- ( (♯ ` ( 1st ` W ) ) e. NN0 -> -. (♯ ` ( 1st ` W ) ) < 0 )
14	13	pm2.21d 622	. . . . . . . . . . 11 |- ( (♯ ` ( 1st ` W ) ) e. NN0 -> ( (♯ ` ( 1st ` W ) ) < 0 -> ( 1st ` W ) = (/) ) )
15	12, 14	sylbird 170	. . . . . . . . . 10 |- ( (♯ ` ( 1st ` W ) ) e. NN0 -> ( ( 0 ... (♯ ` ( 1st ` W ) ) ) = (/) -> ( 1st ` W ) = (/) ) )
16	15	com12 30	. . . . . . . . 9 |- ( ( 0 ... (♯ ` ( 1st ` W ) ) ) = (/) -> ( (♯ ` ( 1st ` W ) ) e. NN0 -> ( 1st ` W ) = (/) ) )
17	16	adantl 277	. . . . . . . 8 |- ( ( ( 2nd ` W ) = (/) /\ ( 0 ... (♯ ` ( 1st ` W ) ) ) = (/) ) -> ( (♯ ` ( 1st ` W ) ) e. NN0 -> ( 1st ` W ) = (/) ) )
18		lencl 11075	. . . . . . . 8 |- ( ( 1st ` W ) e. Word dom (iEdg ` G ) -> (♯ ` ( 1st ` W ) ) e. NN0 )
19	17, 18	impel 280	. . . . . . 7 |- ( ( ( ( 2nd ` W ) = (/) /\ ( 0 ... (♯ ` ( 1st ` W ) ) ) = (/) ) /\ ( 1st ` W ) e. Word dom (iEdg ` G ) ) -> ( 1st ` W ) = (/) )
20		simpll 527	. . . . . . 7 |- ( ( ( ( 2nd ` W ) = (/) /\ ( 0 ... (♯ ` ( 1st ` W ) ) ) = (/) ) /\ ( 1st ` W ) e. Word dom (iEdg ` G ) ) -> ( 2nd ` W ) = (/) )
21	19, 20	jca 306	. . . . . 6 |- ( ( ( ( 2nd ` W ) = (/) /\ ( 0 ... (♯ ` ( 1st ` W ) ) ) = (/) ) /\ ( 1st ` W ) e. Word dom (iEdg ` G ) ) -> ( ( 1st ` W ) = (/) /\ ( 2nd ` W ) = (/) ) )
22	21	ex 115	. . . . 5 |- ( ( ( 2nd ` W ) = (/) /\ ( 0 ... (♯ ` ( 1st ` W ) ) ) = (/) ) -> ( ( 1st ` W ) e. Word dom (iEdg ` G ) -> ( ( 1st ` W ) = (/) /\ ( 2nd ` W ) = (/) ) ) )
23	8, 22	biimtrdi 163	. . . 4 |- ( (Vtx ` G ) = (/) -> ( ( 2nd ` W ) : ( 0 ... (♯ ` ( 1st ` W ) ) ) --> (Vtx ` G ) -> ( ( 1st ` W ) e. Word dom (iEdg ` G ) -> ( ( 1st ` W ) = (/) /\ ( 2nd ` W ) = (/) ) ) ) )
24	23	impcomd 255	. . 3 |- ( (Vtx ` G ) = (/) -> ( ( ( 1st ` W ) e. Word dom (iEdg ` G ) /\ ( 2nd ` W ) : ( 0 ... (♯ ` ( 1st ` W ) ) ) --> (Vtx ` G ) ) -> ( ( 1st ` W ) = (/) /\ ( 2nd ` W ) = (/) ) ) )
25	5, 24	syl5 32	. 2 |- ( (Vtx ` G ) = (/) -> ( ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) -> ( ( 1st ` W ) = (/) /\ ( 2nd ` W ) = (/) ) ) )
26		wlkcprim 16061	. 2 |- ( W e. (Walks ` G ) -> ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) )
27	25, 26	impel 280	1 |- ( ( (Vtx ` G ) = (/) /\ W e. (Walks ` G ) ) -> ( ( 1st ` W ) = (/) /\ ( 2nd ` W ) = (/) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   (/)c0 3491   class class class wbr 4083   dom cdm 4719   -->wf 5314   ` cfv 5318  (class class class)co 6001   1stc1st 6284   2ndc2nd 6285   0cc0 7999   < clt 8181   NN0cn0 9369   ZZcz 9446   ...cfz 10204  ♯chash 10997  Word cword 11071  Vtxcvtx 15813  iEdgciedg 15814  Walkscwlks 16030

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-er 6680  df-map 6797  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-z 9447  df-dec 9579  df-uz 9723  df-fz 10205  df-fzo 10339  df-ihash 10998  df-word 11072  df-ndx 13035  df-slot 13036  df-base 13038  df-edgf 15806  df-vtx 15815  df-iedg 15816  df-wlks 16031

This theorem is referenced by:  g0wlk0  16081