Theorem usgr1vr 26147

Description: A simple graph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 21-Mar-2021.) (Proof shortened by AV, 2-Apr-2021.)

Assertion

Ref	Expression
usgr1vr	|- ( ( A e. X /\ (Vtx ` G ) = { A } ) -> ( G e. USGraph -> (iEdg ` G ) = (/) ) )

Proof of Theorem usgr1vr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgruhgr 26078	. . . . 5 |- ( G e. USGraph -> G e. UHGraph )
2	1	adantl 482	. . . 4 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> G e. UHGraph )
3		fveq2 6191	. . . . . 6 |- ( (Vtx ` G ) = { A } -> ( # ` (Vtx ` G ) ) = ( # ` { A } ) )
4		hashsng 13159	. . . . . 6 |- ( A e. X -> ( # ` { A } ) = 1 )
5	3, 4	sylan9eqr 2678	. . . . 5 |- ( ( A e. X /\ (Vtx ` G ) = { A } ) -> ( # ` (Vtx ` G ) ) = 1 )
6	5	adantr 481	. . . 4 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> ( # ` (Vtx ` G ) ) = 1 )
7		eqid 2622	. . . . . . 7 |- (Vtx ` G ) = (Vtx ` G )
8		eqid 2622	. . . . . . 7 |- (iEdg ` G ) = (iEdg ` G )
9	7, 8	usgrislfuspgr 26079	. . . . . 6 |- ( G e. USGraph <-> ( G e. USPGraph /\ (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) } ) )
10	9	simprbi 480	. . . . 5 |- ( G e. USGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) } )
11	10	adantl 482	. . . 4 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) } )
12		eqid 2622	. . . . 5 |- { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) } = { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) }
13	7, 8, 12	lfuhgr1v0e 26146	. . . 4 |- ( ( G e. UHGraph /\ ( # ` (Vtx ` G ) ) = 1 /\ (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ~P (Vtx ` G ) | 2 <_ ( # ` x ) } ) -> (Edg ` G ) = (/) )
14	2, 6, 11, 13	syl3anc 1326	. . 3 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> (Edg ` G ) = (/) )
15		uhgriedg0edg0 26022	. . . . 5 |- ( G e. UHGraph -> ( (Edg ` G ) = (/) <-> (iEdg ` G ) = (/) ) )
16	1, 15	syl 17	. . . 4 |- ( G e. USGraph -> ( (Edg ` G ) = (/) <-> (iEdg ` G ) = (/) ) )
17	16	adantl 482	. . 3 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> ( (Edg ` G ) = (/) <-> (iEdg ` G ) = (/) ) )
18	14, 17	mpbid 222	. 2 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> (iEdg ` G ) = (/) )
19	18	ex 450	1 |- ( ( A e. X /\ (Vtx ` G ) = { A } ) -> ( G e. USGraph -> (iEdg ` G ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888   1c1 9937   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   USPGraph cuspgr 26043   USGraph cusgr 26044

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-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-rmo 2920  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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  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-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-uspgr 26045  df-usgr 26046

This theorem is referenced by:  usgr1v  26148  usgr1v0e  26218