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Theorem cusgra1v 25181
Description: A graph with one vertex (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
Assertion
Ref Expression
cusgra1v  |-  { A } ComplUSGrph  (/)

Proof of Theorem cusgra1v
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 4660 . . . 4  |-  { A }  e.  _V
2 usgra0 25089 . . . 4  |-  ( { A }  e.  _V  ->  { A } USGrph  (/) )
31, 2mp1i 13 . . 3  |-  ( A  e.  _V  ->  { A } USGrph 
(/) )
4 ral0 3903 . . . 4  |-  A. n  e.  (/)  { n ,  A }  e.  ran  (/)
5 sneq 4007 . . . . . . . 8  |-  ( k  =  A  ->  { k }  =  { A } )
65difeq2d 3584 . . . . . . 7  |-  ( k  =  A  ->  ( { A }  \  {
k } )  =  ( { A }  \  { A } ) )
7 difid 3864 . . . . . . 7  |-  ( { A }  \  { A } )  =  (/)
86, 7syl6eq 2480 . . . . . 6  |-  ( k  =  A  ->  ( { A }  \  {
k } )  =  (/) )
9 preq2 4078 . . . . . . 7  |-  ( k  =  A  ->  { n ,  k }  =  { n ,  A } )
109eleq1d 2492 . . . . . 6  |-  ( k  =  A  ->  ( { n ,  k }  e.  ran  (/)  <->  { n ,  A }  e.  ran  (/) ) )
118, 10raleqbidv 3040 . . . . 5  |-  ( k  =  A  ->  ( A. n  e.  ( { A }  \  {
k } ) { n ,  k }  e.  ran  (/)  <->  A. n  e.  (/)  { n ,  A }  e.  ran  (/) ) )
1211ralsng 4032 . . . 4  |-  ( A  e.  _V  ->  ( A. k  e.  { A } A. n  e.  ( { A }  \  { k } ) { n ,  k }  e.  ran  (/)  <->  A. n  e.  (/)  { n ,  A }  e.  ran  (/) ) )
134, 12mpbiri 237 . . 3  |-  ( A  e.  _V  ->  A. k  e.  { A } A. n  e.  ( { A }  \  { k } ) { n ,  k }  e.  ran  (/) )
14 0ex 4554 . . . 4  |-  (/)  e.  _V
15 iscusgra 25176 . . . 4  |-  ( ( { A }  e.  _V  /\  (/)  e.  _V )  ->  ( { A } ComplUSGrph  (/)  <->  ( { A } USGrph  (/)  /\  A. k  e.  { A } A. n  e.  ( { A }  \  { k } ) { n ,  k }  e.  ran  (/) ) ) )
161, 14, 15mp2an 677 . . 3  |-  ( { A } ComplUSGrph  (/)  <->  ( { A } USGrph 
(/)  /\  A. k  e.  { A } A. n  e.  ( { A }  \  { k } ) { n ,  k }  e.  ran  (/) ) )
173, 13, 16sylanbrc 669 . 2  |-  ( A  e.  _V  ->  { A } ComplUSGrph  (/) )
18 snprc 4061 . . 3  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
19 cusgra0v 25180 . . . 4  |-  (/) ComplUSGrph  (/)
20 breq1 4424 . . . 4  |-  ( { A }  =  (/)  ->  ( { A } ComplUSGrph  (/)  <->  (/) ComplUSGrph  (/) ) )
2119, 20mpbiri 237 . . 3  |-  ( { A }  =  (/)  ->  { A } ComplUSGrph  (/) )
2218, 21sylbi 199 . 2  |-  ( -.  A  e.  _V  ->  { A } ComplUSGrph  (/) )
2317, 22pm2.61i 168 1  |-  { A } ComplUSGrph  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869   A.wral 2776   _Vcvv 3082    \ cdif 3434   (/)c0 3762   {csn 3997   {cpr 3999   class class class wbr 4421   ran crn 4852   USGrph cusg 25049   ComplUSGrph ccusgra 25138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-opab 4481  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-usgra 25052  df-cusgra 25141
This theorem is referenced by: (None)
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