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Metamath Proof Explorer

Theorem usgrnloop

Description: In a simple graph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017) (Proof shortened by Alexander van der Vekens, 20-Mar-2018) (Revised by AV, 17-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

Ref Expression
Hypothesis usgrnloopv.e 
|- E = ( iEdg ` G )
Assertion usgrnloop 
|- ( G e. USGraph -> ( E. x e. dom E ( E ` x ) = { M , N } -> M =/= N ) )

Proof

Step Hyp Ref Expression
1 usgrnloopv.e 
 |-  E = ( iEdg ` G )
2 usgrumgr 
 |-  ( G e. USGraph -> G e. UMGraph )
3 1 umgrnloop 
 |-  ( G e. UMGraph -> ( E. x e. dom E ( E ` x ) = { M , N } -> M =/= N ) )
4 2 3 syl 
 |-  ( G e. USGraph -> ( E. x e. dom E ( E ` x ) = { M , N } -> M =/= N ) )