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Theorem uvtx01vtx 25299
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 4528 . . . . 5  |-  (/)  e.  _V
2 isuvtx 25295 . . . . 5  |-  ( ( V  e.  _V  /\  (/) 
e.  _V )  ->  ( V UnivVertex  (/) )  =  {
x  e.  V  |  A. k  e.  ( V  \  { x }
) { k ,  x }  e.  ran  (/)
} )
31, 2mpan2 685 . . . 4  |-  ( V  e.  _V  ->  ( V UnivVertex  (/) )  =  {
x  e.  V  |  A. k  e.  ( V  \  { x }
) { k ,  x }  e.  ran  (/)
} )
43neeq1d 2702 . . 3  |-  ( V  e.  _V  ->  (
( V UnivVertex  (/) )  =/=  (/)  <->  { x  e.  V  |  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/) }  =/=  (/) ) )
5 rabn0 3755 . . . 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 2762 . . . 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 3726 . . . . . . . . . 10  |-  -.  {
k ,  x }  e.  (/)
9 rn0 5092 . . . . . . . . . . 11  |-  ran  (/)  =  (/)
109eleq2i 2541 . . . . . . . . . 10  |-  ( { k ,  x }  e.  ran  (/)  <->  { k ,  x }  e.  (/) )
118, 10mtbir 306 . . . . . . . . 9  |-  -.  {
k ,  x }  e.  ran  (/)
1211ralf0 3867 . . . . . . . 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 718 . . . . . 6  |-  ( V  e.  _V  ->  (
( x  e.  V  /\  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/) )  <->  ( x  e.  V  /\  ( V  \  { x }
)  =  (/) ) ) )
15 ssdif0 3741 . . . . . . . . 9  |-  ( V 
C_  { x }  <->  ( V  \  { x } )  =  (/) )
1615a1i 11 . . . . . . . 8  |-  ( V  e.  _V  ->  ( V  C_  { x }  <->  ( V  \  { x } )  =  (/) ) )
1716bicomd 206 . . . . . . 7  |-  ( V  e.  _V  ->  (
( V  \  {
x } )  =  (/) 
<->  V  C_  { x } ) )
1817anbi2d 718 . . . . . 6  |-  ( V  e.  _V  ->  (
( x  e.  V  /\  ( V  \  {
x } )  =  (/) )  <->  ( x  e.  V  /\  V  C_  { x } ) ) )
19 sssn 4122 . . . . . . . . 9  |-  ( V 
C_  { x }  <->  ( V  =  (/)  \/  V  =  { x } ) )
2019anbi2i 708 . . . . . . . 8  |-  ( ( x  e.  V  /\  V  C_  { x }
)  <->  ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) ) )
21 n0i 3727 . . . . . . . . . . . 12  |-  ( x  e.  V  ->  -.  V  =  (/) )
2221pm2.21d 109 . . . . . . . . . . 11  |-  ( x  e.  V  ->  ( V  =  (/)  ->  V  =  { x } ) )
2322imp 436 . . . . . . . . . 10  |-  ( ( x  e.  V  /\  V  =  (/) )  ->  V  =  { x } )
24 simpr 468 . . . . . . . . . 10  |-  ( ( x  e.  V  /\  V  =  { x } )  ->  V  =  { x } )
2523, 24jaodan 802 . . . . . . . . 9  |-  ( ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) )  ->  V  =  { x } )
26 ssnid 3989 . . . . . . . . . . 11  |-  x  e. 
{ x }
27 eleq2 2538 . . . . . . . . . . 11  |-  ( V  =  { x }  ->  ( x  e.  V  <->  x  e.  { x }
) )
2826, 27mpbiri 241 . . . . . . . . . 10  |-  ( V  =  { x }  ->  x  e.  V )
29 olc 391 . . . . . . . . . 10  |-  ( V  =  { x }  ->  ( V  =  (/)  \/  V  =  { x } ) )
3028, 29jca 541 . . . . . . . . 9  |-  ( V  =  { x }  ->  ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) ) )
3125, 30impbii 192 . . . . . . . 8  |-  ( ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) )  <->  V  =  {
x } )
3220, 31bitri 257 . . . . . . 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 287 . . . . 5  |-  ( V  e.  _V  ->  (
( x  e.  V  /\  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/) )  <->  V  =  { x } ) )
3534exbidv 1776 . . . 4  |-  ( V  e.  _V  ->  ( E. x ( x  e.  V  /\  A. k  e.  ( V  \  {
x } ) { k ,  x }  e.  ran  (/) )  <->  E. x  V  =  { x } ) )
367, 35syl5bb 265 . . 3  |-  ( V  e.  _V  ->  ( E. x  e.  V  A. k  e.  ( V  \  { x }
) { k ,  x }  e.  ran  (/)  <->  E. x  V  =  {
x } ) )
374, 6, 363bitrd 287 . 2  |-  ( V  e.  _V  ->  (
( V UnivVertex  (/) )  =/=  (/)  <->  E. x  V  =  { x } ) )
38 id 22 . . . . . . 7  |-  ( -.  V  e.  _V  ->  -.  V  e.  _V )
3938intnanrd 931 . . . . . 6  |-  ( -.  V  e.  _V  ->  -.  ( V  e.  _V  /\  (/)  e.  _V ) )
40 df-uvtx 25229 . . . . . . 7  |- UnivVertex  =  ( v  e.  _V , 
e  e.  _V  |->  { n  e.  v  | 
A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e } )
4140mpt2ndm0 6529 . . . . . 6  |-  ( -.  ( V  e.  _V  /\  (/)  e.  _V )  -> 
( V UnivVertex  (/) )  =  (/) )
4239, 41syl 17 . . . . 5  |-  ( -.  V  e.  _V  ->  ( V UnivVertex  (/) )  =  (/) )
4342notnotd 128 . . . 4  |-  ( -.  V  e.  _V  ->  -. 
-.  ( V UnivVertex  (/) )  =  (/) )
44 df-ne 2643 . . . 4  |-  ( ( V UnivVertex  (/) )  =/=  (/)  <->  -.  ( V UnivVertex  (/) )  =  (/) )
4543, 44sylnibr 312 . . 3  |-  ( -.  V  e.  _V  ->  -.  ( V UnivVertex  (/) )  =/=  (/) )
46 snex 4641 . . . . . 6  |-  { x }  e.  _V
47 eleq1 2537 . . . . . 6  |-  ( V  =  { x }  ->  ( V  e.  _V  <->  { x }  e.  _V ) )
4846, 47mpbiri 241 . . . . 5  |-  ( V  =  { x }  ->  V  e.  _V )
4948exlimiv 1784 . . . 4  |-  ( E. x  V  =  {
x }  ->  V  e.  _V )
5049con3i 142 . . 3  |-  ( -.  V  e.  _V  ->  -. 
E. x  V  =  { x } )
5145, 502falsed 358 . 2  |-  ( -.  V  e.  _V  ->  ( ( V UnivVertex  (/) )  =/=  (/) 
<->  E. x  V  =  { x } ) )
5237, 51pm2.61i 169 1  |-  ( ( V UnivVertex  (/) )  =/=  (/)  <->  E. x  V  =  { x } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    \ cdif 3387    C_ wss 3390   (/)c0 3722   {csn 3959   {cpr 3961   ran crn 4840  (class class class)co 6308   UnivVertex cuvtx 25226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-uvtx 25229
This theorem is referenced by: (None)
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