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Theorem uhgra0v 24714
Description: The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
Assertion
Ref Expression
uhgra0v  |-  ( (/) UHGrph  E  <-> 
E  =  (/) )

Proof of Theorem uhgra0v
StepHypRef Expression
1 uhgrav 24700 . . 3  |-  ( (/) UHGrph  E  ->  ( (/)  e.  _V  /\  E  e.  _V )
)
2 isuhgra 24702 . . . 4  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( (/) UHGrph  E 
<->  E : dom  E --> ( ~P (/)  \  { (/) } ) ) )
3 eqid 2402 . . . . . 6  |-  dom  E  =  dom  E
4 pw0 4118 . . . . . . . 8  |-  ~P (/)  =  { (/)
}
54difeq1i 3556 . . . . . . 7  |-  ( ~P (/)  \  { (/) } )  =  ( { (/) } 
\  { (/) } )
6 difid 3839 . . . . . . 7  |-  ( {
(/) }  \  { (/) } )  =  (/)
75, 6eqtri 2431 . . . . . 6  |-  ( ~P (/)  \  { (/) } )  =  (/)
83, 7feq23i 5707 . . . . 5  |-  ( E : dom  E --> ( ~P (/)  \  { (/) } )  <-> 
E : dom  E --> (/) )
9 f00 5749 . . . . . 6  |-  ( E : dom  E --> (/)  <->  ( E  =  (/)  /\  dom  E  =  (/) ) )
109simplbi 458 . . . . 5  |-  ( E : dom  E --> (/)  ->  E  =  (/) )
118, 10sylbi 195 . . . 4  |-  ( E : dom  E --> ( ~P (/)  \  { (/) } )  ->  E  =  (/) )
122, 11syl6bi 228 . . 3  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( (/) UHGrph  E  ->  E  =  (/) ) )
131, 12mpcom 34 . 2  |-  ( (/) UHGrph  E  ->  E  =  (/) )
14 0ex 4525 . . . 4  |-  (/)  e.  _V
15 uhgra0 24713 . . . 4  |-  ( (/)  e.  _V  ->  (/) UHGrph  (/) )
1614, 15ax-mp 5 . . 3  |-  (/) UHGrph  (/)
17 breq2 4398 . . 3  |-  ( E  =  (/)  ->  ( (/) UHGrph  E  <->  (/) UHGrph  (/) ) )
1816, 17mpbiri 233 . 2  |-  ( E  =  (/)  ->  (/) UHGrph  E )
1913, 18impbii 188 1  |-  ( (/) UHGrph  E  <-> 
E  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3058    \ cdif 3410   (/)c0 3737   ~Pcpw 3954   {csn 3971   class class class wbr 4394   dom cdm 4822   -->wf 5564   UHGrph cuhg 24694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-fun 5570  df-fn 5571  df-f 5572  df-uhgra 24696
This theorem is referenced by: (None)
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