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Theorem uvtx01vtx 25206
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 4553 . . . . 5  |-  (/)  e.  _V
2 isuvtx 25202 . . . . 5  |-  ( ( V  e.  _V  /\  (/) 
e.  _V )  ->  ( V UnivVertex  (/) )  =  {
x  e.  V  |  A. k  e.  ( V  \  { x }
) { k ,  x }  e.  ran  (/)
} )
31, 2mpan2 675 . . . 4  |-  ( V  e.  _V  ->  ( V UnivVertex  (/) )  =  {
x  e.  V  |  A. k  e.  ( V  \  { x }
) { k ,  x }  e.  ran  (/)
} )
43neeq1d 2701 . . 3  |-  ( V  e.  _V  ->  (
( V UnivVertex  (/) )  =/=  (/)  <->  { x  e.  V  |  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/) }  =/=  (/) ) )
5 rabn0 3782 . . . 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 2781 . . . 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 3765 . . . . . . . . . 10  |-  -.  {
k ,  x }  e.  (/)
9 rn0 5102 . . . . . . . . . . 11  |-  ran  (/)  =  (/)
109eleq2i 2500 . . . . . . . . . 10  |-  ( { k ,  x }  e.  ran  (/)  <->  { k ,  x }  e.  (/) )
118, 10mtbir 300 . . . . . . . . 9  |-  -.  {
k ,  x }  e.  ran  (/)
1211ralf0 3904 . . . . . . . 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 708 . . . . . 6  |-  ( V  e.  _V  ->  (
( x  e.  V  /\  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/) )  <->  ( x  e.  V  /\  ( V  \  { x }
)  =  (/) ) ) )
15 ssdif0 3851 . . . . . . . . 9  |-  ( V 
C_  { x }  <->  ( V  \  { x } )  =  (/) )
1615a1i 11 . . . . . . . 8  |-  ( V  e.  _V  ->  ( V  C_  { x }  <->  ( V  \  { x } )  =  (/) ) )
1716bicomd 204 . . . . . . 7  |-  ( V  e.  _V  ->  (
( V  \  {
x } )  =  (/) 
<->  V  C_  { x } ) )
1817anbi2d 708 . . . . . 6  |-  ( V  e.  _V  ->  (
( x  e.  V  /\  ( V  \  {
x } )  =  (/) )  <->  ( x  e.  V  /\  V  C_  { x } ) ) )
19 sssn 4155 . . . . . . . . 9  |-  ( V 
C_  { x }  <->  ( V  =  (/)  \/  V  =  { x } ) )
2019anbi2i 698 . . . . . . . 8  |-  ( ( x  e.  V  /\  V  C_  { x }
)  <->  ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) ) )
21 n0i 3766 . . . . . . . . . . . 12  |-  ( x  e.  V  ->  -.  V  =  (/) )
2221pm2.21d 109 . . . . . . . . . . 11  |-  ( x  e.  V  ->  ( V  =  (/)  ->  V  =  { x } ) )
2322imp 430 . . . . . . . . . 10  |-  ( ( x  e.  V  /\  V  =  (/) )  ->  V  =  { x } )
24 simpr 462 . . . . . . . . . 10  |-  ( ( x  e.  V  /\  V  =  { x } )  ->  V  =  { x } )
2523, 24jaodan 792 . . . . . . . . 9  |-  ( ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) )  ->  V  =  { x } )
26 ssnid 4025 . . . . . . . . . . 11  |-  x  e. 
{ x }
27 eleq2 2495 . . . . . . . . . . 11  |-  ( V  =  { x }  ->  ( x  e.  V  <->  x  e.  { x }
) )
2826, 27mpbiri 236 . . . . . . . . . 10  |-  ( V  =  { x }  ->  x  e.  V )
29 olc 385 . . . . . . . . . 10  |-  ( V  =  { x }  ->  ( V  =  (/)  \/  V  =  { x } ) )
3028, 29jca 534 . . . . . . . . 9  |-  ( V  =  { x }  ->  ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) ) )
3125, 30impbii 190 . . . . . . . 8  |-  ( ( x  e.  V  /\  ( V  =  (/)  \/  V  =  { x } ) )  <->  V  =  {
x } )
3220, 31bitri 252 . . . . . . 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 282 . . . . 5  |-  ( V  e.  _V  ->  (
( x  e.  V  /\  A. k  e.  ( V  \  { x } ) { k ,  x }  e.  ran  (/) )  <->  V  =  { x } ) )
3534exbidv 1758 . . . 4  |-  ( V  e.  _V  ->  ( E. x ( x  e.  V  /\  A. k  e.  ( V  \  {
x } ) { k ,  x }  e.  ran  (/) )  <->  E. x  V  =  { x } ) )
367, 35syl5bb 260 . . 3  |-  ( V  e.  _V  ->  ( E. x  e.  V  A. k  e.  ( V  \  { x }
) { k ,  x }  e.  ran  (/)  <->  E. x  V  =  {
x } ) )
374, 6, 363bitrd 282 . 2  |-  ( V  e.  _V  ->  (
( V UnivVertex  (/) )  =/=  (/)  <->  E. x  V  =  { x } ) )
38 id 23 . . . . . . 7  |-  ( -.  V  e.  _V  ->  -.  V  e.  _V )
3938intnanrd 925 . . . . . 6  |-  ( -.  V  e.  _V  ->  -.  ( V  e.  _V  /\  (/)  e.  _V ) )
40 df-uvtx 25136 . . . . . . 7  |- UnivVertex  =  ( v  e.  _V , 
e  e.  _V  |->  { n  e.  v  | 
A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e } )
4140mpt2ndm0 6521 . . . . . 6  |-  ( -.  ( V  e.  _V  /\  (/)  e.  _V )  -> 
( V UnivVertex  (/) )  =  (/) )
4239, 41syl 17 . . . . 5  |-  ( -.  V  e.  _V  ->  ( V UnivVertex  (/) )  =  (/) )
43 notnot 292 . . . . 5  |-  ( ( V UnivVertex  (/) )  =  (/)  <->  -.  -.  ( V UnivVertex  (/) )  =  (/) )
4442, 43sylib 199 . . . 4  |-  ( -.  V  e.  _V  ->  -. 
-.  ( V UnivVertex  (/) )  =  (/) )
45 df-ne 2620 . . . 4  |-  ( ( V UnivVertex  (/) )  =/=  (/)  <->  -.  ( V UnivVertex  (/) )  =  (/) )
4644, 45sylnibr 306 . . 3  |-  ( -.  V  e.  _V  ->  -.  ( V UnivVertex  (/) )  =/=  (/) )
47 snex 4659 . . . . . 6  |-  { x }  e.  _V
48 eleq1 2494 . . . . . 6  |-  ( V  =  { x }  ->  ( V  e.  _V  <->  { x }  e.  _V ) )
4947, 48mpbiri 236 . . . . 5  |-  ( V  =  { x }  ->  V  e.  _V )
5049exlimiv 1766 . . . 4  |-  ( E. x  V  =  {
x }  ->  V  e.  _V )
5150con3i 140 . . 3  |-  ( -.  V  e.  _V  ->  -. 
E. x  V  =  { x } )
5246, 512falsed 352 . 2  |-  ( -.  V  e.  _V  ->  ( ( V UnivVertex  (/) )  =/=  (/) 
<->  E. x  V  =  { x } ) )
5337, 52pm2.61i 167 1  |-  ( ( V UnivVertex  (/) )  =/=  (/)  <->  E. x  V  =  { x } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776   {crab 2779   _Vcvv 3081    \ cdif 3433    C_ wss 3436   (/)c0 3761   {csn 3996   {cpr 3998   ran crn 4851  (class class class)co 6302   UnivVertex cuvtx 25133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-iota 5562  df-fun 5600  df-fv 5606  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-uvtx 25136
This theorem is referenced by: (None)
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