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Theorem uhgra0v 23986
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 23972 . . 3  |-  ( (/) UHGrph  E  ->  ( (/)  e.  _V  /\  E  e.  _V )
)
2 isuhgra 23974 . . . 4  |-  ( (
(/)  e.  _V  /\  E  e.  _V )  ->  ( (/) UHGrph  E 
<->  E : dom  E --> ( ~P (/)  \  { (/) } ) ) )
3 eqid 2467 . . . . . 6  |-  dom  E  =  dom  E
4 pw0 4174 . . . . . . . 8  |-  ~P (/)  =  { (/)
}
54difeq1i 3618 . . . . . . 7  |-  ( ~P (/)  \  { (/) } )  =  ( { (/) } 
\  { (/) } )
6 difid 3895 . . . . . . 7  |-  ( {
(/) }  \  { (/) } )  =  (/)
75, 6eqtri 2496 . . . . . 6  |-  ( ~P (/)  \  { (/) } )  =  (/)
83, 7feq23i 5723 . . . . 5  |-  ( E : dom  E --> ( ~P (/)  \  { (/) } )  <-> 
E : dom  E --> (/) )
9 f00 5765 . . . . . 6  |-  ( E : dom  E --> (/)  <->  ( E  =  (/)  /\  dom  E  =  (/) ) )
109simplbi 460 . . . . 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 36 . 2  |-  ( (/) UHGrph  E  ->  E  =  (/) )
14 0ex 4577 . . . 4  |-  (/)  e.  _V
15 uhgra0 23985 . . . 4  |-  ( (/)  e.  _V  ->  (/) UHGrph  (/) )
1614, 15ax-mp 5 . . 3  |-  (/) UHGrph  (/)
17 breq2 4451 . . 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 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    \ cdif 3473   (/)c0 3785   ~Pcpw 4010   {csn 4027   class class class wbr 4447   dom cdm 4999   -->wf 5582   UHGrph cuhg 23966
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  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-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  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-fun 5588  df-fn 5589  df-f 5590  df-uhgra 23968
This theorem is referenced by: (None)
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