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Theorem uhgraedgrnv 32702
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 24441 . . 3  |-  ( V UHGrph  E  ->  E : dom  E --> ( ~P V  \  { (/) } ) )
2 df-f 5513 . . . 4  |-  ( E : dom  E --> ( ~P V  \  { (/) } )  <->  ( E  Fn  dom  E  /\  ran  E  C_  ( ~P V  \  { (/) } ) ) )
3 ssel2 3425 . . . . . . 7  |-  ( ( ran  E  C_  ( ~P V  \  { (/) } )  /\  F  e. 
ran  E )  ->  F  e.  ( ~P V  \  { (/) } ) )
4 eldif 3412 . . . . . . . 8  |-  ( F  e.  ( ~P V  \  { (/) } )  <->  ( F  e.  ~P V  /\  -.  F  e.  { (/) } ) )
5 elpwi 3949 . . . . . . . . . 10  |-  ( F  e.  ~P V  ->  F  C_  V )
65sseld 3429 . . . . . . . . 9  |-  ( F  e.  ~P V  -> 
( N  e.  F  ->  N  e.  V ) )
76adantr 463 . . . . . . . 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 432 . . . . 5  |-  ( ran 
E  C_  ( ~P V  \  { (/) } )  ->  ( F  e. 
ran  E  ->  ( N  e.  F  ->  N  e.  V ) ) )
1110adantl 464 . . . 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 1188 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 367    /\ w3a 971    e. wcel 1836    \ cdif 3399    C_ wss 3402   (/)c0 3724   ~Pcpw 3940   {csn 3957   class class class wbr 4380   dom cdm 4926   ran crn 4927    Fn wfn 5504   -->wf 5505   UHGrph cuhg 24432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-sep 4501  ax-nul 4509  ax-pr 4614
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-ral 2747  df-rex 2748  df-rab 2751  df-v 3049  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-op 3964  df-br 4381  df-opab 4439  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-fun 5511  df-fn 5512  df-f 5513  df-uhgra 24434
This theorem is referenced by: (None)
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