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Theorem uhgra0 24882
Description: The empty graph, with vertices but no edges, is a hypergraph, analogous to umgra0 24898. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
Assertion
Ref Expression
uhgra0  |-  ( V  e.  W  ->  V UHGrph  (/) )

Proof of Theorem uhgra0
StepHypRef Expression
1 f0 5781 . . 3  |-  (/) : (/) --> ( ~P V  \  { (/)
} )
2 dm0 5068 . . . 4  |-  dom  (/)  =  (/)
32feq2i 5739 . . 3  |-  ( (/) : dom  (/) --> ( ~P V  \  { (/) } )  <->  (/) : (/) --> ( ~P V  \  { (/)
} ) )
41, 3mpbir 212 . 2  |-  (/) : dom  (/) --> ( ~P V  \  { (/)
} )
5 0ex 4557 . . 3  |-  (/)  e.  _V
6 isuhgra 24871 . . 3  |-  ( ( V  e.  W  /\  (/) 
e.  _V )  ->  ( V UHGrph 
(/) 
<->  (/) : dom  (/) --> ( ~P V  \  { (/) } ) ) )
75, 6mpan2 675 . 2  |-  ( V  e.  W  ->  ( V UHGrph 
(/) 
<->  (/) : dom  (/) --> ( ~P V  \  { (/) } ) ) )
84, 7mpbiri 236 1  |-  ( V  e.  W  ->  V UHGrph  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    e. wcel 1870   _Vcvv 3087    \ cdif 3439   (/)c0 3767   ~Pcpw 3985   {csn 4002   class class class wbr 4426   dom cdm 4854   -->wf 5597   UHGrph cuhg 24863
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-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  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-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  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-fun 5603  df-fn 5604  df-f 5605  df-uhgra 24865
This theorem is referenced by:  uhgra0v  24883
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