Theorem 1conngr 27054

Description: A graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 15-Feb-2021.)

Assertion

Ref	Expression
1conngr	|- ( ( G e. W /\ (Vtx ` G ) = { N } ) -> G e. ConnGraph )

Proof of Theorem 1conngr

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

Step	Hyp	Ref	Expression
1		snidg 4206	. . . . . . . . . 10 |- ( N e. _V -> N e. { N } )
2	1	adantr 481	. . . . . . . . 9 |- ( ( N e. _V /\ ( G e. W /\ (Vtx ` G ) = { N } ) ) -> N e. { N } )
3		eleq2 2690	. . . . . . . . . 10 |- ( (Vtx ` G ) = { N } -> ( N e. (Vtx ` G ) <-> N e. { N } ) )
4	3	ad2antll 765	. . . . . . . . 9 |- ( ( N e. _V /\ ( G e. W /\ (Vtx ` G ) = { N } ) ) -> ( N e. (Vtx ` G ) <-> N e. { N } ) )
5	2, 4	mpbird 247	. . . . . . . 8 |- ( ( N e. _V /\ ( G e. W /\ (Vtx ` G ) = { N } ) ) -> N e. (Vtx ` G ) )
6		eqid 2622	. . . . . . . . 9 |- (Vtx ` G ) = (Vtx ` G )
7	6	0pthonv 26990	. . . . . . . 8 |- ( N e. (Vtx ` G ) -> E. f E. p f ( N (PathsOn ` G ) N ) p )
8	5, 7	syl 17	. . . . . . 7 |- ( ( N e. _V /\ ( G e. W /\ (Vtx ` G ) = { N } ) ) -> E. f E. p f ( N (PathsOn ` G ) N ) p )
9		oveq2 6658	. . . . . . . . . . 11 |- ( n = N -> ( N (PathsOn ` G ) n ) = ( N (PathsOn ` G ) N ) )
10	9	breqd 4664	. . . . . . . . . 10 |- ( n = N -> ( f ( N (PathsOn ` G ) n ) p <-> f ( N (PathsOn ` G ) N ) p ) )
11	10	2exbidv 1852	. . . . . . . . 9 |- ( n = N -> ( E. f E. p f ( N (PathsOn ` G ) n ) p <-> E. f E. p f ( N (PathsOn ` G ) N ) p ) )
12	11	ralsng 4218	. . . . . . . 8 |- ( N e. _V -> ( A. n e. { N } E. f E. p f ( N (PathsOn ` G ) n ) p <-> E. f E. p f ( N (PathsOn ` G ) N ) p ) )
13	12	adantr 481	. . . . . . 7 |- ( ( N e. _V /\ ( G e. W /\ (Vtx ` G ) = { N } ) ) -> ( A. n e. { N } E. f E. p f ( N (PathsOn ` G ) n ) p <-> E. f E. p f ( N (PathsOn ` G ) N ) p ) )
14	8, 13	mpbird 247	. . . . . 6 |- ( ( N e. _V /\ ( G e. W /\ (Vtx ` G ) = { N } ) ) -> A. n e. { N } E. f E. p f ( N (PathsOn ` G ) n ) p )
15		oveq1 6657	. . . . . . . . . . 11 |- ( k = N -> ( k (PathsOn ` G ) n ) = ( N (PathsOn ` G ) n ) )
16	15	breqd 4664	. . . . . . . . . 10 |- ( k = N -> ( f ( k (PathsOn ` G ) n ) p <-> f ( N (PathsOn ` G ) n ) p ) )
17	16	2exbidv 1852	. . . . . . . . 9 |- ( k = N -> ( E. f E. p f ( k (PathsOn ` G ) n ) p <-> E. f E. p f ( N (PathsOn ` G ) n ) p ) )
18	17	ralbidv 2986	. . . . . . . 8 |- ( k = N -> ( A. n e. { N } E. f E. p f ( k (PathsOn ` G ) n ) p <-> A. n e. { N } E. f E. p f ( N (PathsOn ` G ) n ) p ) )
19	18	ralsng 4218	. . . . . . 7 |- ( N e. _V -> ( A. k e. { N } A. n e. { N } E. f E. p f ( k (PathsOn ` G ) n ) p <-> A. n e. { N } E. f E. p f ( N (PathsOn ` G ) n ) p ) )
20	19	adantr 481	. . . . . 6 |- ( ( N e. _V /\ ( G e. W /\ (Vtx ` G ) = { N } ) ) -> ( A. k e. { N } A. n e. { N } E. f E. p f ( k (PathsOn ` G ) n ) p <-> A. n e. { N } E. f E. p f ( N (PathsOn ` G ) n ) p ) )
21	14, 20	mpbird 247	. . . . 5 |- ( ( N e. _V /\ ( G e. W /\ (Vtx ` G ) = { N } ) ) -> A. k e. { N } A. n e. { N } E. f E. p f ( k (PathsOn ` G ) n ) p )
22		id 22	. . . . . . 7 |- ( (Vtx ` G ) = { N } -> (Vtx ` G ) = { N } )
23		raleq 3138	. . . . . . 7 |- ( (Vtx ` G ) = { N } -> ( A. n e. (Vtx ` G ) E. f E. p f ( k (PathsOn ` G ) n ) p <-> A. n e. { N } E. f E. p f ( k (PathsOn ` G ) n ) p ) )
24	22, 23	raleqbidv 3152	. . . . . 6 |- ( (Vtx ` G ) = { N } -> ( A. k e. (Vtx ` G ) A. n e. (Vtx ` G ) E. f E. p f ( k (PathsOn ` G ) n ) p <-> A. k e. { N } A. n e. { N } E. f E. p f ( k (PathsOn ` G ) n ) p ) )
25	24	ad2antll 765	. . . . 5 |- ( ( N e. _V /\ ( G e. W /\ (Vtx ` G ) = { N } ) ) -> ( A. k e. (Vtx ` G ) A. n e. (Vtx ` G ) E. f E. p f ( k (PathsOn ` G ) n ) p <-> A. k e. { N } A. n e. { N } E. f E. p f ( k (PathsOn ` G ) n ) p ) )
26	21, 25	mpbird 247	. . . 4 |- ( ( N e. _V /\ ( G e. W /\ (Vtx ` G ) = { N } ) ) -> A. k e. (Vtx ` G ) A. n e. (Vtx ` G ) E. f E. p f ( k (PathsOn ` G ) n ) p )
27	6	isconngr 27049	. . . . 5 |- ( G e. W -> ( G e. ConnGraph <-> A. k e. (Vtx ` G ) A. n e. (Vtx ` G ) E. f E. p f ( k (PathsOn ` G ) n ) p ) )
28	27	ad2antrl 764	. . . 4 |- ( ( N e. _V /\ ( G e. W /\ (Vtx ` G ) = { N } ) ) -> ( G e. ConnGraph <-> A. k e. (Vtx ` G ) A. n e. (Vtx ` G ) E. f E. p f ( k (PathsOn ` G ) n ) p ) )
29	26, 28	mpbird 247	. . 3 |- ( ( N e. _V /\ ( G e. W /\ (Vtx ` G ) = { N } ) ) -> G e. ConnGraph )
30	29	ex 450	. 2 |- ( N e. _V -> ( ( G e. W /\ (Vtx ` G ) = { N } ) -> G e. ConnGraph ) )
31		snprc 4253	. . 3 |- ( -. N e. _V <-> { N } = (/) )
32		eqeq2 2633	. . . . 5 |- ( { N } = (/) -> ( (Vtx ` G ) = { N } <-> (Vtx ` G ) = (/) ) )
33	32	anbi2d 740	. . . 4 |- ( { N } = (/) -> ( ( G e. W /\ (Vtx ` G ) = { N } ) <-> ( G e. W /\ (Vtx ` G ) = (/) ) ) )
34		0vconngr 27053	. . . 4 |- ( ( G e. W /\ (Vtx ` G ) = (/) ) -> G e. ConnGraph )
35	33, 34	syl6bi 243	. . 3 |- ( { N } = (/) -> ( ( G e. W /\ (Vtx ` G ) = { N } ) -> G e. ConnGraph ) )
36	31, 35	sylbi 207	. 2 |- ( -. N e. _V -> ( ( G e. W /\ (Vtx ` G ) = { N } ) -> G e. ConnGraph ) )
37	30, 36	pm2.61i 176	1 |- ( ( G e. W /\ (Vtx ` G ) = { N } ) -> G e. ConnGraph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   _Vcvv 3200   (/)c0 3915   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  PathsOncpthson 26610  ConnGraphcconngr 27046

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-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-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-pthson 26614  df-conngr 27047

This theorem is referenced by: (None)