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Theorem uvtx0 24195
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 24192 . . 3  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( (/) UnivVertex  E )  =  { n  e.  (/)  |  A. k  e.  ( (/)  \  { n } ) { k ,  n }  e.  ran  E } )
2 ral0 3932 . . . . 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 3807 . . . 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 212 . . 3  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  { n  e.  (/)  |  A. k  e.  ( (/)  \  { n } ) { k ,  n }  e.  ran  E }  =  (/) )
61, 5eqtrd 2508 . 2  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( (/) UnivVertex  E )  =  (/) )
7 df-uvtx 24126 . . 3  |- UnivVertex  =  ( v  e.  _V , 
e  e.  _V  |->  { n  e.  v  | 
A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e } )
87mpt2ndm0 6500 . 2  |-  ( -.  ( (/)  e.  _V  /\  E  e.  _V )  ->  ( (/) UnivVertex  E )  =  (/) )
96, 8pm2.61i 164 1  |-  ( (/) UnivVertex  E
)  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    \ cdif 3473   (/)c0 3785   {csn 4027   {cpr 4029   ran crn 5000  (class class class)co 6284   UnivVertex cuvtx 24123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-uvtx 24126
This theorem is referenced by: (None)
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