Theorem usgr1vr 16472

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 16412	. . . . . . . 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 2238	. . . . . . . 8 |- (Vtx ` G ) = (Vtx ` G )
4		eqid 2238	. . . . . . . 8 |- (Edg ` G ) = (Edg ` G )
5	3, 4	upgredg 16368	. . . . . . 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 541	. . . . . . . . . . . 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 550	. . . . . . . . . . . 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 2317	. . . . . . . . . . 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 3726	. . . . . . . . . . 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 542	. . . . . . . . . . . 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 2317	. . . . . . . . . . 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 3726	. . . . . . . . . . 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 2274	. . . . . . . . 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 548	. . . . . . . . . 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 540	. . . . . . . . . . 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 2316	. . . . . . . . . 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 16428	. . . . . . . . . . 11 |- ( ( G e. USGraph /\ { p , q } e. (Edg ` G ) ) -> p =/= q )
22	21	neneqd 2441	. . . . . . . . . 10 |- ( ( G e. USGraph /\ { p , q } e. (Edg ` G ) ) -> -. p = q )
23	17, 20, 22	syl2anc 415	. . . . . . . . 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 1422	. . . . . . . 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 2676	. . . . . 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 1421	. . . 4 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> -. e e. (Edg ` G ) )
29	28	eq0rdv 3571	. . 3 |- ( ( ( A e. X /\ (Vtx ` G ) = { A } ) /\ G e. USGraph ) -> (Edg ` G ) = (/) )
30		usgruhgr 16413	. . . . 5 |- ( G e. USGraph -> G e. UHGraph )
31		uhgriedg0edg0 16359	. . . . 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 1402   F. wfal 1407   e. wcel 2209   E.wrex 2529   (/)c0 3520   {csn 3708   {cpr 3709   ` cfv 5375  Vtxcvtx 16236  iEdgciedg 16237  Edgcedg 16281  UHGraphcuhgr 16291  UPGraphcupgr 16315  USGraphcusgr 16378

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-1o 6681  df-2o 6682  df-er 6801  df-en 7017  df-sub 8493  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-dec 9761  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-edg 16282  df-uhgrm 16293  df-upgren 16317  df-umgren 16318  df-uspgren 16379  df-usgren 16380

This theorem is referenced by: (None)