Metamath Proof Explorer

Theorem uhgr0edg0rgrb

Description: A hypergraph is 0-regular iff it has no edges. (Contributed by Alexander van der Vekens, 12-Jul-2018) (Revised by AV, 24-Dec-2020)

Ref Expression
Assertion uhgr0edg0rgrb 
|- ( G e. UHGraph -> ( G RegGraph 0 <-> ( Edg ` G ) = (/) ) )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
2 eqid 
 |-  ( Edg ` G ) = ( Edg ` G )
3 1 2 uhgrvd00 
 |-  ( G e. UHGraph -> ( A. v e. ( Vtx ` G ) ( ( VtxDeg ` G ) ` v ) = 0 -> ( Edg ` G ) = (/) ) )
4 3 com12 
 |-  ( A. v e. ( Vtx ` G ) ( ( VtxDeg ` G ) ` v ) = 0 -> ( G e. UHGraph -> ( Edg ` G ) = (/) ) )
5 4 adantl 
 |-  ( ( 0 e. NN0* /\ A. v e. ( Vtx ` G ) ( ( VtxDeg ` G ) ` v ) = 0 ) -> ( G e. UHGraph -> ( Edg ` G ) = (/) ) )
6 eqid 
 |-  ( VtxDeg ` G ) = ( VtxDeg ` G )
7 1 6 rgrprop 
 |-  ( G RegGraph 0 -> ( 0 e. NN0* /\ A. v e. ( Vtx ` G ) ( ( VtxDeg ` G ) ` v ) = 0 ) )
8 5 7 syl11 
 |-  ( G e. UHGraph -> ( G RegGraph 0 -> ( Edg ` G ) = (/) ) )
9 uhgr0edg0rgr 
 |-  ( ( G e. UHGraph /\ ( Edg ` G ) = (/) ) -> G RegGraph 0 )
10 9 ex 
 |-  ( G e. UHGraph -> ( ( Edg ` G ) = (/) -> G RegGraph 0 ) )
11 8 10 impbid 
 |-  ( G e. UHGraph -> ( G RegGraph 0 <-> ( Edg ` G ) = (/) ) )