MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cusgra1v Structured version   Unicode version

Theorem cusgra1v 24125
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 4683 . . . 4  |-  { A }  e.  _V
2 usgra0 24034 . . . 4  |-  ( { A }  e.  _V  ->  { A } USGrph  (/) )
31, 2mp1i 12 . . 3  |-  ( A  e.  _V  ->  { A } USGrph 
(/) )
4 ral0 3927 . . . 4  |-  A. n  e.  (/)  { n ,  A }  e.  ran  (/)
5 sneq 4032 . . . . . . . 8  |-  ( k  =  A  ->  { k }  =  { A } )
65difeq2d 3617 . . . . . . 7  |-  ( k  =  A  ->  ( { A }  \  {
k } )  =  ( { A }  \  { A } ) )
7 difid 3890 . . . . . . 7  |-  ( { A }  \  { A } )  =  (/)
86, 7syl6eq 2519 . . . . . 6  |-  ( k  =  A  ->  ( { A }  \  {
k } )  =  (/) )
9 preq2 4102 . . . . . . 7  |-  ( k  =  A  ->  { n ,  k }  =  { n ,  A } )
109eleq1d 2531 . . . . . 6  |-  ( k  =  A  ->  ( { n ,  k }  e.  ran  (/)  <->  { n ,  A }  e.  ran  (/) ) )
118, 10raleqbidv 3067 . . . . 5  |-  ( k  =  A  ->  ( A. n  e.  ( { A }  \  {
k } ) { n ,  k }  e.  ran  (/)  <->  A. n  e.  (/)  { n ,  A }  e.  ran  (/) ) )
1211ralsng 4057 . . . 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 4572 . . . 4  |-  (/)  e.  _V
15 iscusgra 24120 . . . 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 672 . . 3  |-  ( { A } ComplUSGrph  (/)  <->  ( { A } USGrph 
(/)  /\  A. k  e.  { A } A. n  e.  ( { A }  \  { k } ) { n ,  k }  e.  ran  (/) ) )
173, 13, 16sylanbrc 664 . 2  |-  ( A  e.  _V  ->  { A } ComplUSGrph  (/) )
18 snprc 4086 . . 3  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
19 cusgra0v 24124 . . . 4  |-  (/) ComplUSGrph  (/)
20 breq1 4445 . . . 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 369    = wceq 1374    e. wcel 1762   A.wral 2809   _Vcvv 3108    \ cdif 3468   (/)c0 3780   {csn 4022   {cpr 4024   class class class wbr 4442   ran crn 4995   USGrph cusg 23995   ComplUSGrph ccusgra 24082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-usgra 23998  df-cusgra 24085
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator