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Theorem uvtx01vtx 23319
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 4419 . . . . 5  |-  (/)  e.  _V
2 isuvtx 23315 . . . . 5  |-  ( ( V  e.  _V  /\  (/) 
e.  _V )  ->  ( V UnivVertex  (/) )  =  {
x  e.  V  |  A. k  e.  ( V  \  { x }
) { k ,  x }  e.  ran  (/)
} )
31, 2mpan2 666 . . . 4  |-  ( V  e.  _V  ->  ( V UnivVertex  (/) )  =  {
x  e.  V  |  A. k  e.  ( V  \  { x }
) { k ,  x }  e.  ran  (/)
} )
43neeq1d 2619 . . 3  |-  ( V  e.  _V  ->  (
( V UnivVertex  (/) )  =/=  (/)  <->  { x  e.  V  |  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/) }  =/=  (/) ) )
5 rabn0 3654 . . . 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 2719 . . . 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 3638 . . . . . . . . . 10  |-  -.  {
k ,  x }  e.  (/)
9 rn0 5087 . . . . . . . . . . 11  |-  ran  (/)  =  (/)
109eleq2i 2505 . . . . . . . . . 10  |-  ( { k ,  x }  e.  ran  (/)  <->  { k ,  x }  e.  (/) )
118, 10mtbir 299 . . . . . . . . 9  |-  -.  {
k ,  x }  e.  ran  (/)
1211ralf0 3783 . . . . . . . 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 698 . . . . . 6  |-  ( V  e.  _V  ->  (
( x  e.  V  /\  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/) )  <->  ( x  e.  V  /\  ( V  \  { x }
)  =  (/) ) ) )
15 ssdif0 3734 . . . . . . . . 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 698 . . . . . 6  |-  ( V  e.  _V  ->  (
( x  e.  V  /\  ( V  \  {
x } )  =  (/) )  <->  ( x  e.  V  /\  V  C_  { x } ) ) )
19 sssn 4028 . . . . . . . . 9  |-  ( V 
C_  { x }  <->  ( V  =  (/)  \/  V  =  { x } ) )
2019anbi2i 689 . . . . . . . 8  |-  ( ( x  e.  V  /\  V  C_  { x }
)  <->  ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) ) )
21 n0i 3639 . . . . . . . . . . . 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 458 . . . . . . . . . 10  |-  ( ( x  e.  V  /\  V  =  { x } )  ->  V  =  { x } )
2523, 24jaodan 778 . . . . . . . . 9  |-  ( ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) )  ->  V  =  { x } )
26 ssnid 3903 . . . . . . . . . . 11  |-  x  e. 
{ x }
27 eleq2 2502 . . . . . . . . . . 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 529 . . . . . . . . 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 1685 . . . 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 903 . . . . . 6  |-  ( -.  V  e.  _V  ->  -.  ( V  e.  _V  /\  (/)  e.  _V ) )
40 df-uvtx 23253 . . . . . . 7  |- UnivVertex  =  ( v  e.  _V , 
e  e.  _V  |->  { n  e.  v  | 
A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e } )
4140mpt2ndm0 6738 . . . . . 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 2606 . . . 4  |-  ( ( V UnivVertex  (/) )  =/=  (/)  <->  -.  ( V UnivVertex  (/) )  =  (/) )
4644, 45sylnibr 305 . . 3  |-  ( -.  V  e.  _V  ->  -.  ( V UnivVertex  (/) )  =/=  (/) )
47 snex 4530 . . . . . 6  |-  { x }  e.  _V
48 eleq1 2501 . . . . . 6  |-  ( V  =  { x }  ->  ( V  e.  _V  <->  { x }  e.  _V ) )
4947, 48mpbiri 233 . . . . 5  |-  ( V  =  { x }  ->  V  e.  _V )
5049exlimiv 1693 . . . 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 1364   E.wex 1591    e. wcel 1761    =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717   _Vcvv 2970    \ cdif 3322    C_ wss 3325   (/)c0 3634   {csn 3874   {cpr 3876   ran crn 4837  (class class class)co 6090   UnivVertex cuvtx 23250
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-uvtx 23253
This theorem is referenced by: (None)
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