Theorem wlkv0 16524

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 2238	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
2	1	wlkf 16485	. . . 4 |- ( ( 1st ` W ) (Walks ` G ) ( 2nd ` W ) -> ( 1st ` W ) e. Word dom (iEdg ` G ) )
3		eqid 2238	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
4	3	wlkp 16489	. . . 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 5513	. . . . . 6 |- ( (Vtx ` G ) = (/) -> ( ( 2nd ` W ) : ( 0 ... (♯ ` ( 1st ` W ) ) ) --> (Vtx ` G ) <-> ( 2nd ` W ) : ( 0 ... (♯ ` ( 1st ` W ) ) ) --> (/) ) )
7		f00 5579	. . . . . 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 9634	. . . . . . . . . . . 12 |- 0 e. ZZ
10		nn0z 9643	. . . . . . . . . . . 12 |- ( (♯ ` ( 1st ` W ) ) e. NN0 -> (♯ ` ( 1st ` W ) ) e. ZZ )
11		fzn 10425	. . . . . . . . . . . 12 |- ( ( 0 e. ZZ /\ (♯ ` ( 1st ` W ) ) e. ZZ ) -> ( (♯ ` ( 1st ` W ) ) < 0 <-> ( 0 ... (♯ ` ( 1st ` W ) ) ) = (/) ) )
12	9, 10, 11	sylancr 418	. . . . . . . . . . 11 |- ( (♯ ` ( 1st ` W ) ) e. NN0 -> ( (♯ ` ( 1st ` W ) ) < 0 <-> ( 0 ... (♯ ` ( 1st ` W ) ) ) = (/) ) )
13		nn0nlt0 9568	. . . . . . . . . . . 12 |- ( (♯ ` ( 1st ` W ) ) e. NN0 -> -. (♯ ` ( 1st ` W ) ) < 0 )
14	13	pm2.21d 628	. . . . . . . . . . 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 11286	. . . . . . . 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 531	. . . . . . 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 16505	. 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 1402   e. wcel 2209   (/)c0 3520   class class class wbr 4125   dom cdm 4769   -->wf 5368   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363   0cc0 8169   < clt 8350   NN0cn0 9542   ZZcz 9623   ...cfz 10390  ♯chash 11192  Word cword 11282  Vtxcvtx 16167  iEdgciedg 16168  Walkscwlks 16472

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-ifp 991  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-map 6914  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-z 9624  df-dec 9757  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-wlks 16473

This theorem is referenced by:  g0wlk0  16525