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Metamath Proof Explorer

Theorem umgrnloop

Description: In a multigraph, there is no loop, i.e. no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017) (Revised by AV, 11-Dec-2020)

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

Proof

Step Hyp Ref Expression
1 umgrnloopv.e 
 |-  E = ( iEdg ` G )
2 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
3 1 2 umgredgprv 
 |-  ( ( G e. UMGraph /\ x e. dom E ) -> ( ( E ` x ) = { M , N } -> ( M e. ( Vtx ` G ) /\ N e. ( Vtx ` G ) ) ) )
4 3 imp 
 |-  ( ( ( G e. UMGraph /\ x e. dom E ) /\ ( E ` x ) = { M , N } ) -> ( M e. ( Vtx ` G ) /\ N e. ( Vtx ` G ) ) )
5 1 umgrnloopv 
 |-  ( ( G e. UMGraph /\ M e. ( Vtx ` G ) ) -> ( ( E ` x ) = { M , N } -> M =/= N ) )
6 5 ex 
 |-  ( G e. UMGraph -> ( M e. ( Vtx ` G ) -> ( ( E ` x ) = { M , N } -> M =/= N ) ) )
7 6 com23 
 |-  ( G e. UMGraph -> ( ( E ` x ) = { M , N } -> ( M e. ( Vtx ` G ) -> M =/= N ) ) )
8 7 adantr 
 |-  ( ( G e. UMGraph /\ x e. dom E ) -> ( ( E ` x ) = { M , N } -> ( M e. ( Vtx ` G ) -> M =/= N ) ) )
9 8 imp 
 |-  ( ( ( G e. UMGraph /\ x e. dom E ) /\ ( E ` x ) = { M , N } ) -> ( M e. ( Vtx ` G ) -> M =/= N ) )
10 9 com12 
 |-  ( M e. ( Vtx ` G ) -> ( ( ( G e. UMGraph /\ x e. dom E ) /\ ( E ` x ) = { M , N } ) -> M =/= N ) )
11 10 adantr 
 |-  ( ( M e. ( Vtx ` G ) /\ N e. ( Vtx ` G ) ) -> ( ( ( G e. UMGraph /\ x e. dom E ) /\ ( E ` x ) = { M , N } ) -> M =/= N ) )
12 4 11 mpcom 
 |-  ( ( ( G e. UMGraph /\ x e. dom E ) /\ ( E ` x ) = { M , N } ) -> M =/= N )
13 12 rexlimdva2 
 |-  ( G e. UMGraph -> ( E. x e. dom E ( E ` x ) = { M , N } -> M =/= N ) )