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Theorem elusuhgra 25144
Description: An undirected simple graph without loops is an undirected hypergraph. (Contributed by AV, 9-Jan-2020.)
Assertion
Ref Expression
elusuhgra  |-  ( G  e. USGrph  ->  G  e. UHGrph  )

Proof of Theorem elusuhgra
StepHypRef Expression
1 relusgra 25111 . . 3  |-  Rel USGrph
2 1st2nd 6866 . . 3  |-  ( ( Rel USGrph  /\  G  e. USGrph  )  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
31, 2mpan 681 . 2  |-  ( G  e. USGrph  ->  G  =  <. ( 1st `  G ) ,  ( 2nd `  G
) >. )
4 usisuhgra 25143 . . . 4  |-  ( ( 1st `  G ) USGrph 
( 2nd `  G
)  ->  ( 1st `  G ) UHGrph  ( 2nd `  G ) )
54a1i 11 . . 3  |-  ( G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  ->  (
( 1st `  G
) USGrph  ( 2nd `  G
)  ->  ( 1st `  G ) UHGrph  ( 2nd `  G ) ) )
6 eleq1 2528 . . . 4  |-  ( G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  ->  ( G  e. USGrph  <->  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  e. USGrph  )
)
7 df-br 4417 . . . 4  |-  ( ( 1st `  G ) USGrph 
( 2nd `  G
)  <->  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  e. USGrph  )
86, 7syl6bbr 271 . . 3  |-  ( G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  ->  ( G  e. USGrph  <->  ( 1st `  G
) USGrph  ( 2nd `  G
) ) )
9 eleq1 2528 . . . 4  |-  ( G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  ->  ( G  e. UHGrph  <->  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  e. UHGrph  )
)
10 df-br 4417 . . . 4  |-  ( ( 1st `  G ) UHGrph 
( 2nd `  G
)  <->  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  e. UHGrph  )
119, 10syl6bbr 271 . . 3  |-  ( G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  ->  ( G  e. UHGrph  <->  ( 1st `  G
) UHGrph  ( 2nd `  G
) ) )
125, 8, 113imtr4d 276 . 2  |-  ( G  =  <. ( 1st `  G
) ,  ( 2nd `  G ) >.  ->  ( G  e. USGrph  ->  G  e. UHGrph  ) )
133, 12mpcom 37 1  |-  ( G  e. USGrph  ->  G  e. UHGrph  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1455    e. wcel 1898   <.cop 3986   class class class wbr 4416   Rel wrel 4858   ` cfv 5601   1stc1st 6818   2ndc2nd 6819   UHGrph cuhg 25066   USGrph cusg 25106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-i2m1 9633  ax-1ne0 9634  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-1st 6820  df-2nd 6821  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-2 10696  df-uhgra 25068  df-umgra 25089  df-uslgra 25108  df-usgra 25109
This theorem is referenced by:  0eusgraiff0rgracl  25718  usgsizedg  39980  usgo0s0  40020  usgo1s0  40027
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