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Theorem uvtx01vtx 23579
Description: If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. This theorem could have been stated  ( ( V UnivVertex  (/) )  =/=  (/)  <->  ( # `  V
)  =  1 ), but a lot of auxiliary theorems would have been needed. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
uvtx01vtx  |-  ( ( V UnivVertex  (/) )  =/=  (/)  <->  E. x  V  =  { x } )
Distinct variable group:    x, V

Proof of Theorem uvtx01vtx
Dummy variables  e 
k  n  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4533 . . . . 5  |-  (/)  e.  _V
2 isuvtx 23575 . . . . 5  |-  ( ( V  e.  _V  /\  (/) 
e.  _V )  ->  ( V UnivVertex  (/) )  =  {
x  e.  V  |  A. k  e.  ( V  \  { x }
) { k ,  x }  e.  ran  (/)
} )
31, 2mpan2 671 . . . 4  |-  ( V  e.  _V  ->  ( V UnivVertex  (/) )  =  {
x  e.  V  |  A. k  e.  ( V  \  { x }
) { k ,  x }  e.  ran  (/)
} )
43neeq1d 2729 . . 3  |-  ( V  e.  _V  ->  (
( V UnivVertex  (/) )  =/=  (/)  <->  { x  e.  V  |  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/) }  =/=  (/) ) )
5 rabn0 3768 . . . 4  |-  ( { x  e.  V  |  A. k  e.  ( V  \  { x }
) { k ,  x }  e.  ran  (/)
}  =/=  (/)  <->  E. x  e.  V  A. k  e.  ( V  \  {
x } ) { k ,  x }  e.  ran  (/) )
65a1i 11 . . 3  |-  ( V  e.  _V  ->  ( { x  e.  V  |  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/) }  =/=  (/)  <->  E. x  e.  V  A. k  e.  ( V  \  {
x } ) { k ,  x }  e.  ran  (/) ) )
7 df-rex 2805 . . . 4  |-  ( E. x  e.  V  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/)  <->  E. x
( x  e.  V  /\  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/) ) )
8 noel 3752 . . . . . . . . . 10  |-  -.  {
k ,  x }  e.  (/)
9 rn0 5202 . . . . . . . . . . 11  |-  ran  (/)  =  (/)
109eleq2i 2532 . . . . . . . . . 10  |-  ( { k ,  x }  e.  ran  (/)  <->  { k ,  x }  e.  (/) )
118, 10mtbir 299 . . . . . . . . 9  |-  -.  {
k ,  x }  e.  ran  (/)
1211ralf0 3897 . . . . . . . 8  |-  ( A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/)  <->  ( V  \  { x } )  =  (/) )
1312a1i 11 . . . . . . 7  |-  ( V  e.  _V  ->  ( A. k  e.  ( V  \  { x }
) { k ,  x }  e.  ran  (/)  <->  ( V  \  { x } )  =  (/) ) )
1413anbi2d 703 . . . . . 6  |-  ( V  e.  _V  ->  (
( x  e.  V  /\  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/) )  <->  ( x  e.  V  /\  ( V  \  { x }
)  =  (/) ) ) )
15 ssdif0 3848 . . . . . . . . 9  |-  ( V 
C_  { x }  <->  ( V  \  { x } )  =  (/) )
1615a1i 11 . . . . . . . 8  |-  ( V  e.  _V  ->  ( V  C_  { x }  <->  ( V  \  { x } )  =  (/) ) )
1716bicomd 201 . . . . . . 7  |-  ( V  e.  _V  ->  (
( V  \  {
x } )  =  (/) 
<->  V  C_  { x } ) )
1817anbi2d 703 . . . . . 6  |-  ( V  e.  _V  ->  (
( x  e.  V  /\  ( V  \  {
x } )  =  (/) )  <->  ( x  e.  V  /\  V  C_  { x } ) ) )
19 sssn 4142 . . . . . . . . 9  |-  ( V 
C_  { x }  <->  ( V  =  (/)  \/  V  =  { x } ) )
2019anbi2i 694 . . . . . . . 8  |-  ( ( x  e.  V  /\  V  C_  { x }
)  <->  ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) ) )
21 n0i 3753 . . . . . . . . . . . 12  |-  ( x  e.  V  ->  -.  V  =  (/) )
2221pm2.21d 106 . . . . . . . . . . 11  |-  ( x  e.  V  ->  ( V  =  (/)  ->  V  =  { x } ) )
2322imp 429 . . . . . . . . . 10  |-  ( ( x  e.  V  /\  V  =  (/) )  ->  V  =  { x } )
24 simpr 461 . . . . . . . . . 10  |-  ( ( x  e.  V  /\  V  =  { x } )  ->  V  =  { x } )
2523, 24jaodan 783 . . . . . . . . 9  |-  ( ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) )  ->  V  =  { x } )
26 ssnid 4017 . . . . . . . . . . 11  |-  x  e. 
{ x }
27 eleq2 2527 . . . . . . . . . . 11  |-  ( V  =  { x }  ->  ( x  e.  V  <->  x  e.  { x }
) )
2826, 27mpbiri 233 . . . . . . . . . 10  |-  ( V  =  { x }  ->  x  e.  V )
29 olc 384 . . . . . . . . . 10  |-  ( V  =  { x }  ->  ( V  =  (/)  \/  V  =  { x } ) )
3028, 29jca 532 . . . . . . . . 9  |-  ( V  =  { x }  ->  ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) ) )
3125, 30impbii 188 . . . . . . . 8  |-  ( ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) )  <->  V  =  {
x } )
3220, 31bitri 249 . . . . . . 7  |-  ( ( x  e.  V  /\  V  C_  { x }
)  <->  V  =  {
x } )
3332a1i 11 . . . . . 6  |-  ( V  e.  _V  ->  (
( x  e.  V  /\  V  C_  { x } )  <->  V  =  { x } ) )
3414, 18, 333bitrd 279 . . . . 5  |-  ( V  e.  _V  ->  (
( x  e.  V  /\  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/) )  <->  V  =  { x } ) )
3534exbidv 1681 . . . 4  |-  ( V  e.  _V  ->  ( E. x ( x  e.  V  /\  A. k  e.  ( V  \  {
x } ) { k ,  x }  e.  ran  (/) )  <->  E. x  V  =  { x } ) )
367, 35syl5bb 257 . . 3  |-  ( V  e.  _V  ->  ( E. x  e.  V  A. k  e.  ( V  \  { x }
) { k ,  x }  e.  ran  (/)  <->  E. x  V  =  {
x } ) )
374, 6, 363bitrd 279 . 2  |-  ( V  e.  _V  ->  (
( V UnivVertex  (/) )  =/=  (/)  <->  E. x  V  =  { x } ) )
38 id 22 . . . . . . 7  |-  ( -.  V  e.  _V  ->  -.  V  e.  _V )
3938intnanrd 908 . . . . . 6  |-  ( -.  V  e.  _V  ->  -.  ( V  e.  _V  /\  (/)  e.  _V ) )
40 df-uvtx 23513 . . . . . . 7  |- UnivVertex  =  ( v  e.  _V , 
e  e.  _V  |->  { n  e.  v  | 
A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e } )
4140mpt2ndm0 6852 . . . . . 6  |-  ( -.  ( V  e.  _V  /\  (/)  e.  _V )  -> 
( V UnivVertex  (/) )  =  (/) )
4239, 41syl 16 . . . . 5  |-  ( -.  V  e.  _V  ->  ( V UnivVertex  (/) )  =  (/) )
43 notnot 291 . . . . 5  |-  ( ( V UnivVertex  (/) )  =  (/)  <->  -.  -.  ( V UnivVertex  (/) )  =  (/) )
4442, 43sylib 196 . . . 4  |-  ( -.  V  e.  _V  ->  -. 
-.  ( V UnivVertex  (/) )  =  (/) )
45 df-ne 2650 . . . 4  |-  ( ( V UnivVertex  (/) )  =/=  (/)  <->  -.  ( V UnivVertex  (/) )  =  (/) )
4644, 45sylnibr 305 . . 3  |-  ( -.  V  e.  _V  ->  -.  ( V UnivVertex  (/) )  =/=  (/) )
47 snex 4644 . . . . . 6  |-  { x }  e.  _V
48 eleq1 2526 . . . . . 6  |-  ( V  =  { x }  ->  ( V  e.  _V  <->  { x }  e.  _V ) )
4947, 48mpbiri 233 . . . . 5  |-  ( V  =  { x }  ->  V  e.  _V )
5049exlimiv 1689 . . . 4  |-  ( E. x  V  =  {
x }  ->  V  e.  _V )
5150con3i 135 . . 3  |-  ( -.  V  e.  _V  ->  -. 
E. x  V  =  { x } )
5246, 512falsed 351 . 2  |-  ( -.  V  e.  _V  ->  ( ( V UnivVertex  (/) )  =/=  (/) 
<->  E. x  V  =  { x } ) )
5337, 52pm2.61i 164 1  |-  ( ( V UnivVertex  (/) )  =/=  (/)  <->  E. x  V  =  { x } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   {crab 2803   _Vcvv 3078    \ cdif 3436    C_ wss 3439   (/)c0 3748   {csn 3988   {cpr 3990   ran crn 4952  (class class class)co 6203   UnivVertex cuvtx 23510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-uvtx 23513
This theorem is referenced by: (None)
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