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Metamath Proof Explorer

Theorem usgrnloopv

Description: In a simple graph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018) (Revised by AV, 17-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

Ref Expression
Hypothesis usgrnloopv.e 
|- E = ( iEdg ` G )
Assertion usgrnloopv 
|- ( ( G e. USGraph /\ M e. W ) -> ( ( 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 umgrnloopv 
 |-  ( ( G e. UMGraph /\ M e. W ) -> ( ( E ` X ) = { M , N } -> M =/= N ) )
4 2 3 sylan 
 |-  ( ( G e. USGraph /\ M e. W ) -> ( ( E ` X ) = { M , N } -> M =/= N ) )