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Theorem uvtx0 25064
Description: There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
uvtx0  |-  ( (/) UnivVertex  E
)  =  (/)

Proof of Theorem uvtx0
Dummy variables  e 
k  n  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isuvtx 25061 . . 3  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( (/) UnivVertex  E )  =  { n  e.  (/)  |  A. k  e.  ( (/)  \  { n } ) { k ,  n }  e.  ran  E } )
2 ral0 3908 . . . . 5  |-  A. n  e.  (/)  -.  A. k  e.  ( (/)  \  { n } ) { k ,  n }  e.  ran  E
32a1i 11 . . . 4  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  A. n  e.  (/)  -.  A. k  e.  ( (/)  \  { n } ) { k ,  n }  e.  ran  E )
4 rabeq0 3790 . . . 4  |-  ( { n  e.  (/)  |  A. k  e.  ( (/)  \  {
n } ) { k ,  n }  e.  ran  E }  =  (/)  <->  A. n  e.  (/)  -.  A. k  e.  ( (/)  \  {
n } ) { k ,  n }  e.  ran  E )
53, 4sylibr 215 . . 3  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  { n  e.  (/)  |  A. k  e.  ( (/)  \  { n } ) { k ,  n }  e.  ran  E }  =  (/) )
61, 5eqtrd 2470 . 2  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( (/) UnivVertex  E )  =  (/) )
7 df-uvtx 24995 . . 3  |- UnivVertex  =  ( v  e.  _V , 
e  e.  _V  |->  { n  e.  v  | 
A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e } )
87mpt2ndm0 6524 . 2  |-  ( -.  ( (/)  e.  _V  /\  E  e.  _V )  ->  ( (/) UnivVertex  E )  =  (/) )
96, 8pm2.61i 167 1  |-  ( (/) UnivVertex  E
)  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   {crab 2786   _Vcvv 3087    \ cdif 3439   (/)c0 3767   {csn 4002   {cpr 4004   ran crn 4855  (class class class)co 6305   UnivVertex cuvtx 24992
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-uvtx 24995
This theorem is referenced by: (None)
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