Theorem usgr1vr 16292

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 variables e p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		usgrupgr 16232	. . . . . . . 8 |- ( G e. USGraph -> G e. UPGraph )
2	1	adantl 277	. . . . . . 7 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> G e. UPGraph )
3		eqid 2234	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
4		eqid 2234	. . . . . . . 8 |- (Edg ` G ) = (Edg ` G )
5	3, 4	upgredg 16188	. . . . . . 7 |- ( ( G e. UPGraph /\ e e. (Edg ` G ) ) -> E. p e. (Vtx ` G ) E. q e. (Vtx ` G ) e = { p , q } )
6	2, 5	sylan 283	. . . . . 6 |- ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) -> E. p e. (Vtx ` G ) E. q e. (Vtx ` G ) e = { p , q } )
7		simplrl 537	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> p e. (Vtx ` G ) )
8		simp-5r 546	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> (Vtx ` G ) = { A } )
9	7, 8	eleqtrd 2313	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> p e. { A } )
10		elsni 3709	. . . . . . . . . . 11 |- ( p e. { A } -> p = A )
11	9, 10	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> p = A )
12		simplrr 538	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> q e. (Vtx ` G ) )
13	12, 8	eleqtrd 2313	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> q e. { A } )
14		elsni 3709	. . . . . . . . . . 11 |- ( q e. { A } -> q = A )
15	13, 14	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> q = A )
16	11, 15	eqtr4d 2270	. . . . . . . . 9 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> p = q )
17		simp-4r 544	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> G e. USGraph )
18		simpr 110	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> e = { p , q } )
19		simpllr 536	. . . . . . . . . . 11 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> e e. (Edg ` G ) )
20	18, 19	eqeltrrd 2312	. . . . . . . . . 10 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> { p , q } e. (Edg ` G ) )
21	4	usgredgne 16248	. . . . . . . . . . 11 |- ( ( G e. USGraph /\ { p , q } e. (Edg ` G ) ) -> p =/= q )
22	21	neneqd 2435	. . . . . . . . . 10 |- ( ( G e. USGraph /\ { p , q } e. (Edg ` G ) ) -> -. p = q )
23	17, 20, 22	syl2anc 411	. . . . . . . . 9 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> -. p = q )
24	16, 23	pm2.21fal 1418	. . . . . . . 8 |- ( ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) /\ e = { p , q } ) -> F. )
25	24	ex 115	. . . . . . 7 |- ( ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) /\ ( p e. (Vtx ` G ) /\ q e. (Vtx ` G ) ) ) -> ( e = { p , q } -> F. ) )
26	25	rexlimdvva 2670	. . . . . 6 |- ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) -> ( E. p e. (Vtx ` G ) E. q e. (Vtx ` G ) e = { p , q } -> F. ) )
27	6, 26	mpd 13	. . . . 5 |- ( ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) /\ e e. (Edg ` G ) ) -> F. )
28	27	inegd 1417	. . . 4 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> -. e e. (Edg ` G ) )
29	28	eq0rdv 3555	. . 3 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> (Edg ` G ) = (/) )
30		usgruhgr 16233	. . . . 5 |- ( G e. USGraph -> G e. UHGraph )
31		uhgriedg0edg0 16179	. . . . 5 |- ( G e. UHGraph -> ( (Edg ` G ) = (/) <-> (iEdg ` G ) = (/) ) )
32	30, 31	syl 14	. . . 4 |- ( G e. USGraph -> ( (Edg ` G ) = (/) <-> (iEdg ` G ) = (/) ) )
33	32	adantl 277	. . 3 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> ( (Edg ` G ) = (/) <-> (iEdg ` G ) = (/) ) )
34	29, 33	mpbid 147	. 2 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> (iEdg ` G ) = (/) )
35	34	ex 115	1 |- ( ( A e. X /\ (Vtx ` G ) = { A } ) -> ( G e. USGraph -> (iEdg ` G ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   F. wfal 1403   e. wcel 2205   E.wrex 2523   (/)c0 3510   {csn 3691   {cpr 3692   ` cfv 5354  Vtxcvtx 16056  iEdgciedg 16057  Edgcedg 16101  UHGraphcuhgr 16111  UPGraphcupgr 16135  USGraphcusgr 16198

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-cnre 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-1o 6649  df-2o 6650  df-er 6769  df-en 6978  df-sub 8451  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-9 9308  df-n0 9502  df-dec 9716  df-ndx 13236  df-slot 13237  df-base 13239  df-edgf 16049  df-vtx 16058  df-iedg 16059  df-edg 16102  df-uhgrm 16113  df-upgren 16137  df-umgren 16138  df-uspgren 16199  df-usgren 16200

This theorem is referenced by: (None)