Theorem wspthsnonn0vne 26813

Description: If the set of simple paths of length at least 1 between two vertices is not empty, the two vertices must be different. (Contributed by Alexander van der Vekens, 3-Mar-2018.) (Revised by AV, 16-May-2021.)

Assertion

Ref	Expression
wspthsnonn0vne	|- ( ( N e. NN /\ ( X ( N WSPathsNOn G ) Y ) =/= (/) ) -> X =/= Y )

Proof of Theorem wspthsnonn0vne

Dummy variables f p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3931	. . 3 |- ( ( X ( N WSPathsNOn G ) Y ) =/= (/) <-> E. p p e. ( X ( N WSPathsNOn G ) Y ) )
2		eqid 2622	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
3	2	wspthnonp 26744	. . . . 5 |- ( p e. ( X ( N WSPathsNOn G ) Y ) -> ( ( N e. NN0 /\ G e. _V ) /\ ( X e. (Vtx ` G ) /\ Y e. (Vtx ` G ) ) /\ ( p e. ( X ( N WWalksNOn G ) Y ) /\ E. f f ( X (SPathsOn ` G ) Y ) p ) ) )
4	2	wwlknon 26742	. . . . . . . . 9 |- ( ( X e. (Vtx ` G ) /\ Y e. (Vtx ` G ) ) -> ( p e. ( X ( N WWalksNOn G ) Y ) <-> ( p e. ( N WWalksN G ) /\ ( p ` 0 ) = X /\ ( p ` N ) = Y ) ) )
5	4	adantl 482	. . . . . . . 8 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( X e. (Vtx ` G ) /\ Y e. (Vtx ` G ) ) ) -> ( p e. ( X ( N WWalksNOn G ) Y ) <-> ( p e. ( N WWalksN G ) /\ ( p ` 0 ) = X /\ ( p ` N ) = Y ) ) )
6		iswwlksn 26730	. . . . . . . . . . . . . 14 |- ( N e. NN0 -> ( p e. ( N WWalksN G ) <-> ( p e. (WWalks ` G ) /\ ( # ` p ) = ( N + 1 ) ) ) )
7		spthonisspth 26646	. . . . . . . . . . . . . . . . . . 19 |- ( f ( X (SPathsOn ` G ) Y ) p -> f (SPaths ` G ) p )
8		spthispth 26622	. . . . . . . . . . . . . . . . . . 19 |- ( f (SPaths ` G ) p -> f (Paths ` G ) p )
9		pthiswlk 26623	. . . . . . . . . . . . . . . . . . . 20 |- ( f (Paths ` G ) p -> f (Walks ` G ) p )
10		wlklenvm1 26517	. . . . . . . . . . . . . . . . . . . 20 |- ( f (Walks ` G ) p -> ( # ` f ) = ( ( # ` p ) - 1 ) )
11	9, 10	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( f (Paths ` G ) p -> ( # ` f ) = ( ( # ` p ) - 1 ) )
12	7, 8, 11	3syl 18	. . . . . . . . . . . . . . . . . 18 |- ( f ( X (SPathsOn ` G ) Y ) p -> ( # ` f ) = ( ( # ` p ) - 1 ) )
13		oveq1 6657	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` p ) = ( N + 1 ) -> ( ( # ` p ) - 1 ) = ( ( N + 1 ) - 1 ) )
14	13	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` p ) = ( N + 1 ) -> ( ( # ` f ) = ( ( # ` p ) - 1 ) <-> ( # ` f ) = ( ( N + 1 ) - 1 ) ) )
15		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N e. NN /\ ( # ` f ) = ( ( N + 1 ) - 1 ) ) -> ( # ` f ) = ( ( N + 1 ) - 1 ) )
16		nncn 11028	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N e. NN -> N e. CC )
17		pncan1 10454	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N e. CC -> ( ( N + 1 ) - 1 ) = N )
18	16, 17	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N e. NN -> ( ( N + 1 ) - 1 ) = N )
19	18	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N e. NN /\ ( # ` f ) = ( ( N + 1 ) - 1 ) ) -> ( ( N + 1 ) - 1 ) = N )
20	15, 19	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N e. NN /\ ( # ` f ) = ( ( N + 1 ) - 1 ) ) -> ( # ` f ) = N )
21		nnne0 11053	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( N e. NN -> N =/= 0 )
22	21	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N e. NN /\ ( # ` f ) = ( ( N + 1 ) - 1 ) ) -> N =/= 0 )
23	20, 22	eqnetrd 2861	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( N e. NN /\ ( # ` f ) = ( ( N + 1 ) - 1 ) ) -> ( # ` f ) =/= 0 )
24		spthonepeq 26648	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( f ( X (SPathsOn ` G ) Y ) p -> ( X = Y <-> ( # ` f ) = 0 ) )
25	24	necon3bid 2838	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f ( X (SPathsOn ` G ) Y ) p -> ( X =/= Y <-> ( # ` f ) =/= 0 ) )
26	23, 25	syl5ibrcom 237	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( N e. NN /\ ( # ` f ) = ( ( N + 1 ) - 1 ) ) -> ( f ( X (SPathsOn ` G ) Y ) p -> X =/= Y ) )
27	26	expcom 451	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` f ) = ( ( N + 1 ) - 1 ) -> ( N e. NN -> ( f ( X (SPathsOn ` G ) Y ) p -> X =/= Y ) ) )
28	27	com23 86	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` f ) = ( ( N + 1 ) - 1 ) -> ( f ( X (SPathsOn ` G ) Y ) p -> ( N e. NN -> X =/= Y ) ) )
29	14, 28	syl6bi 243	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` p ) = ( N + 1 ) -> ( ( # ` f ) = ( ( # ` p ) - 1 ) -> ( f ( X (SPathsOn ` G ) Y ) p -> ( N e. NN -> X =/= Y ) ) ) )
30	29	com13 88	. . . . . . . . . . . . . . . . . 18 |- ( f ( X (SPathsOn ` G ) Y ) p -> ( ( # ` f ) = ( ( # ` p ) - 1 ) -> ( ( # ` p ) = ( N + 1 ) -> ( N e. NN -> X =/= Y ) ) ) )
31	12, 30	mpd 15	. . . . . . . . . . . . . . . . 17 |- ( f ( X (SPathsOn ` G ) Y ) p -> ( ( # ` p ) = ( N + 1 ) -> ( N e. NN -> X =/= Y ) ) )
32	31	exlimiv 1858	. . . . . . . . . . . . . . . 16 |- ( E. f f ( X (SPathsOn ` G ) Y ) p -> ( ( # ` p ) = ( N + 1 ) -> ( N e. NN -> X =/= Y ) ) )
33	32	com12 32	. . . . . . . . . . . . . . 15 |- ( ( # ` p ) = ( N + 1 ) -> ( E. f f ( X (SPathsOn ` G ) Y ) p -> ( N e. NN -> X =/= Y ) ) )
34	33	adantl 482	. . . . . . . . . . . . . 14 |- ( ( p e. (WWalks ` G ) /\ ( # ` p ) = ( N + 1 ) ) -> ( E. f f ( X (SPathsOn ` G ) Y ) p -> ( N e. NN -> X =/= Y ) ) )
35	6, 34	syl6bi 243	. . . . . . . . . . . . 13 |- ( N e. NN0 -> ( p e. ( N WWalksN G ) -> ( E. f f ( X (SPathsOn ` G ) Y ) p -> ( N e. NN -> X =/= Y ) ) ) )
36	35	adantr 481	. . . . . . . . . . . 12 |- ( ( N e. NN0 /\ G e. _V ) -> ( p e. ( N WWalksN G ) -> ( E. f f ( X (SPathsOn ` G ) Y ) p -> ( N e. NN -> X =/= Y ) ) ) )
37	36	adantr 481	. . . . . . . . . . 11 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( X e. (Vtx ` G ) /\ Y e. (Vtx ` G ) ) ) -> ( p e. ( N WWalksN G ) -> ( E. f f ( X (SPathsOn ` G ) Y ) p -> ( N e. NN -> X =/= Y ) ) ) )
38	37	com12 32	. . . . . . . . . 10 |- ( p e. ( N WWalksN G ) -> ( ( ( N e. NN0 /\ G e. _V ) /\ ( X e. (Vtx ` G ) /\ Y e. (Vtx ` G ) ) ) -> ( E. f f ( X (SPathsOn ` G ) Y ) p -> ( N e. NN -> X =/= Y ) ) ) )
39	38	3ad2ant1 1082	. . . . . . . . 9 |- ( ( p e. ( N WWalksN G ) /\ ( p ` 0 ) = X /\ ( p ` N ) = Y ) -> ( ( ( N e. NN0 /\ G e. _V ) /\ ( X e. (Vtx ` G ) /\ Y e. (Vtx ` G ) ) ) -> ( E. f f ( X (SPathsOn ` G ) Y ) p -> ( N e. NN -> X =/= Y ) ) ) )
40	39	com12 32	. . . . . . . 8 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( X e. (Vtx ` G ) /\ Y e. (Vtx ` G ) ) ) -> ( ( p e. ( N WWalksN G ) /\ ( p ` 0 ) = X /\ ( p ` N ) = Y ) -> ( E. f f ( X (SPathsOn ` G ) Y ) p -> ( N e. NN -> X =/= Y ) ) ) )
41	5, 40	sylbid 230	. . . . . . 7 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( X e. (Vtx ` G ) /\ Y e. (Vtx ` G ) ) ) -> ( p e. ( X ( N WWalksNOn G ) Y ) -> ( E. f f ( X (SPathsOn ` G ) Y ) p -> ( N e. NN -> X =/= Y ) ) ) )
42	41	impd 447	. . . . . 6 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( X e. (Vtx ` G ) /\ Y e. (Vtx ` G ) ) ) -> ( ( p e. ( X ( N WWalksNOn G ) Y ) /\ E. f f ( X (SPathsOn ` G ) Y ) p ) -> ( N e. NN -> X =/= Y ) ) )
43	42	3impia 1261	. . . . 5 |- ( ( ( N e. NN0 /\ G e. _V ) /\ ( X e. (Vtx ` G ) /\ Y e. (Vtx ` G ) ) /\ ( p e. ( X ( N WWalksNOn G ) Y ) /\ E. f f ( X (SPathsOn ` G ) Y ) p ) ) -> ( N e. NN -> X =/= Y ) )
44	3, 43	syl 17	. . . 4 |- ( p e. ( X ( N WSPathsNOn G ) Y ) -> ( N e. NN -> X =/= Y ) )
45	44	exlimiv 1858	. . 3 |- ( E. p p e. ( X ( N WSPathsNOn G ) Y ) -> ( N e. NN -> X =/= Y ) )
46	1, 45	sylbi 207	. 2 |- ( ( X ( N WSPathsNOn G ) Y ) =/= (/) -> ( N e. NN -> X =/= Y ) )
47	46	impcom 446	1 |- ( ( N e. NN /\ ( X ( N WSPathsNOn G ) Y ) =/= (/) ) -> X =/= Y )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   NN0cn0 11292   #chash 13117  Vtxcvtx 25874  Walkscwlks 26492  Pathscpths 26608  SPathscspths 26609  SPathsOncspthson 26611  WWalkscwwlks 26717   WWalksN cwwlksn 26718   WWalksNOn cwwlksnon 26719   WSPathsNOn cwwspthsnon 26721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-spthson 26615  df-wwlksn 26723  df-wwlksnon 26724  df-wspthsnon 26726

This theorem is referenced by:  wspniunwspnon  26819  usgr2wspthons3  26857