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Theorem uhgraedgrnv 30413
Description: An edge of an undirected hypergraph contains only vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
Assertion
Ref Expression
uhgraedgrnv  |-  ( ( V UHGrph  E  /\  F  e. 
ran  E  /\  N  e.  F )  ->  N  e.  V )

Proof of Theorem uhgraedgrnv
StepHypRef Expression
1 uhgraf 23375 . . 3  |-  ( V UHGrph  E  ->  E : dom  E --> ( ~P V  \  { (/) } ) )
2 df-f 5522 . . . 4  |-  ( E : dom  E --> ( ~P V  \  { (/) } )  <->  ( E  Fn  dom  E  /\  ran  E  C_  ( ~P V  \  { (/) } ) ) )
3 ssel2 3451 . . . . . . 7  |-  ( ( ran  E  C_  ( ~P V  \  { (/) } )  /\  F  e. 
ran  E )  ->  F  e.  ( ~P V  \  { (/) } ) )
4 eldif 3438 . . . . . . . 8  |-  ( F  e.  ( ~P V  \  { (/) } )  <->  ( F  e.  ~P V  /\  -.  F  e.  { (/) } ) )
5 elpwi 3969 . . . . . . . . . 10  |-  ( F  e.  ~P V  ->  F  C_  V )
65sseld 3455 . . . . . . . . 9  |-  ( F  e.  ~P V  -> 
( N  e.  F  ->  N  e.  V ) )
76adantr 465 . . . . . . . 8  |-  ( ( F  e.  ~P V  /\  -.  F  e.  { (/)
} )  ->  ( N  e.  F  ->  N  e.  V ) )
84, 7sylbi 195 . . . . . . 7  |-  ( F  e.  ( ~P V  \  { (/) } )  -> 
( N  e.  F  ->  N  e.  V ) )
93, 8syl 16 . . . . . 6  |-  ( ( ran  E  C_  ( ~P V  \  { (/) } )  /\  F  e. 
ran  E )  -> 
( N  e.  F  ->  N  e.  V ) )
109ex 434 . . . . 5  |-  ( ran 
E  C_  ( ~P V  \  { (/) } )  ->  ( F  e. 
ran  E  ->  ( N  e.  F  ->  N  e.  V ) ) )
1110adantl 466 . . . 4  |-  ( ( E  Fn  dom  E  /\  ran  E  C_  ( ~P V  \  { (/) } ) )  ->  ( F  e.  ran  E  -> 
( N  e.  F  ->  N  e.  V ) ) )
122, 11sylbi 195 . . 3  |-  ( E : dom  E --> ( ~P V  \  { (/) } )  ->  ( F  e.  ran  E  ->  ( N  e.  F  ->  N  e.  V ) ) )
131, 12syl 16 . 2  |-  ( V UHGrph  E  ->  ( F  e. 
ran  E  ->  ( N  e.  F  ->  N  e.  V ) ) )
14133imp 1182 1  |-  ( ( V UHGrph  E  /\  F  e. 
ran  E  /\  N  e.  F )  ->  N  e.  V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1758    \ cdif 3425    C_ wss 3428   (/)c0 3737   ~Pcpw 3960   {csn 3977   class class class wbr 4392   dom cdm 4940   ran crn 4941    Fn wfn 5513   -->wf 5514   UHGrph cuhg 23370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-fun 5520  df-fn 5521  df-f 5522  df-uhgra 23371
This theorem is referenced by: (None)
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