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Theorem cusgra1v 24582
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 4603 . . . 4  |-  { A }  e.  _V
2 usgra0 24491 . . . 4  |-  ( { A }  e.  _V  ->  { A } USGrph  (/) )
31, 2mp1i 12 . . 3  |-  ( A  e.  _V  ->  { A } USGrph 
(/) )
4 ral0 3850 . . . 4  |-  A. n  e.  (/)  { n ,  A }  e.  ran  (/)
5 sneq 3954 . . . . . . . 8  |-  ( k  =  A  ->  { k }  =  { A } )
65difeq2d 3536 . . . . . . 7  |-  ( k  =  A  ->  ( { A }  \  {
k } )  =  ( { A }  \  { A } ) )
7 difid 3812 . . . . . . 7  |-  ( { A }  \  { A } )  =  (/)
86, 7syl6eq 2439 . . . . . 6  |-  ( k  =  A  ->  ( { A }  \  {
k } )  =  (/) )
9 preq2 4024 . . . . . . 7  |-  ( k  =  A  ->  { n ,  k }  =  { n ,  A } )
109eleq1d 2451 . . . . . 6  |-  ( k  =  A  ->  ( { n ,  k }  e.  ran  (/)  <->  { n ,  A }  e.  ran  (/) ) )
118, 10raleqbidv 2993 . . . . 5  |-  ( k  =  A  ->  ( A. n  e.  ( { A }  \  {
k } ) { n ,  k }  e.  ran  (/)  <->  A. n  e.  (/)  { n ,  A }  e.  ran  (/) ) )
1211ralsng 3979 . . . 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 233 . . 3  |-  ( A  e.  _V  ->  A. k  e.  { A } A. n  e.  ( { A }  \  { k } ) { n ,  k }  e.  ran  (/) )
14 0ex 4497 . . . 4  |-  (/)  e.  _V
15 iscusgra 24577 . . . 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 670 . . 3  |-  ( { A } ComplUSGrph  (/)  <->  ( { A } USGrph 
(/)  /\  A. k  e.  { A } A. n  e.  ( { A }  \  { k } ) { n ,  k }  e.  ran  (/) ) )
173, 13, 16sylanbrc 662 . 2  |-  ( A  e.  _V  ->  { A } ComplUSGrph  (/) )
18 snprc 4007 . . 3  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
19 cusgra0v 24581 . . . 4  |-  (/) ComplUSGrph  (/)
20 breq1 4370 . . . 4  |-  ( { A }  =  (/)  ->  ( { A } ComplUSGrph  (/)  <->  (/) ComplUSGrph  (/) ) )
2119, 20mpbiri 233 . . 3  |-  ( { A }  =  (/)  ->  { A } ComplUSGrph  (/) )
2218, 21sylbi 195 . 2  |-  ( -.  A  e.  _V  ->  { A } ComplUSGrph  (/) )
2317, 22pm2.61i 164 1  |-  { A } ComplUSGrph  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034    \ cdif 3386   (/)c0 3711   {csn 3944   {cpr 3946   class class class wbr 4367   ran crn 4914   USGrph cusg 24451   ComplUSGrph ccusgra 24539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-usgra 24454  df-cusgra 24542
This theorem is referenced by: (None)
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