Theorem usgr1vr 16067

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 16007	. . . . . . . 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 2229	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
4		eqid 2229	. . . . . . . 8 |- (Edg ` G ) = (Edg ` G )
5	3, 4	upgredg 15963	. . . . . . 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 535	. . . . . . . . . . . 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 544	. . . . . . . . . . . 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 2308	. . . . . . . . . . 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 3684	. . . . . . . . . . 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 536	. . . . . . . . . . . 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 2308	. . . . . . . . . . 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 3684	. . . . . . . . . . 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 2265	. . . . . . . . 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 542	. . . . . . . . . 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 534	. . . . . . . . . . 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 2307	. . . . . . . . . 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 16023	. . . . . . . . . . 11 |- ( ( G e. USGraph /\ { p , q } e. (Edg ` G ) ) -> p =/= q )
22	21	neneqd 2421	. . . . . . . . . 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 1415	. . . . . . . 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 2656	. . . . . 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 1414	. . . 4 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> -. e e. (Edg ` G ) )
29	28	eq0rdv 3536	. . 3 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> (Edg ` G ) = (/) )
30		usgruhgr 16008	. . . . 5 |- ( G e. USGraph -> G e. UHGraph )
31		uhgriedg0edg0 15954	. . . . 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 1395   F. wfal 1400   e. wcel 2200   E.wrex 2509   (/)c0 3491   {csn 3666   {cpr 3667   ` cfv 5321  Vtxcvtx 15834  iEdgciedg 15835  Edgcedg 15879  UHGraphcuhgr 15888  UPGraphcupgr 15912  USGraphcusgr 15973

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-1o 6573  df-2o 6574  df-er 6693  df-en 6901  df-sub 8335  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-n0 9386  df-dec 9595  df-ndx 13056  df-slot 13057  df-base 13059  df-edgf 15827  df-vtx 15836  df-iedg 15837  df-edg 15880  df-uhgrm 15890  df-upgren 15914  df-umgren 15915  df-uspgren 15974  df-usgren 15975

This theorem is referenced by: (None)