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Metamath Proof Explorer

Theorem umgrnloopv

Description: In a multigraph, 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, 11-Dec-2020)

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

Proof

Step Hyp Ref Expression
1 umgrnloopv.e 
 |-  E = ( iEdg ` G )
2 prnzg 
 |-  ( M e. W -> { M , N } =/= (/) )
3 2 adantl 
 |-  ( ( ( E ` X ) = { M , N } /\ M e. W ) -> { M , N } =/= (/) )
4 neeq1 
 |-  ( ( E ` X ) = { M , N } -> ( ( E ` X ) =/= (/) <-> { M , N } =/= (/) ) )
5 4 adantr 
 |-  ( ( ( E ` X ) = { M , N } /\ M e. W ) -> ( ( E ` X ) =/= (/) <-> { M , N } =/= (/) ) )
6 3 5 mpbird 
 |-  ( ( ( E ` X ) = { M , N } /\ M e. W ) -> ( E ` X ) =/= (/) )
7 fvfundmfvn0 
 |-  ( ( E ` X ) =/= (/) -> ( X e. dom E /\ Fun ( E |` { X } ) ) )
8 6 7 syl 
 |-  ( ( ( E ` X ) = { M , N } /\ M e. W ) -> ( X e. dom E /\ Fun ( E |` { X } ) ) )
9 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
10 9 1 umgredg2 
 |-  ( ( G e. UMGraph /\ X e. dom E ) -> ( # ` ( E ` X ) ) = 2 )
11 fveqeq2 
 |-  ( ( E ` X ) = { M , N } -> ( ( # ` ( E ` X ) ) = 2 <-> ( # ` { M , N } ) = 2 ) )
12 eqid 
 |-  { M , N } = { M , N }
13 12 hashprdifel 
 |-  ( ( # ` { M , N } ) = 2 -> ( M e. { M , N } /\ N e. { M , N } /\ M =/= N ) )
14 13 simp3d 
 |-  ( ( # ` { M , N } ) = 2 -> M =/= N )
15 11 14 biimtrdi 
 |-  ( ( E ` X ) = { M , N } -> ( ( # ` ( E ` X ) ) = 2 -> M =/= N ) )
16 15 adantr 
 |-  ( ( ( E ` X ) = { M , N } /\ M e. W ) -> ( ( # ` ( E ` X ) ) = 2 -> M =/= N ) )
17 10 16 syl5com 
 |-  ( ( G e. UMGraph /\ X e. dom E ) -> ( ( ( E ` X ) = { M , N } /\ M e. W ) -> M =/= N ) )
18 17 expcom 
 |-  ( X e. dom E -> ( G e. UMGraph -> ( ( ( E ` X ) = { M , N } /\ M e. W ) -> M =/= N ) ) )
19 18 com23 
 |-  ( X e. dom E -> ( ( ( E ` X ) = { M , N } /\ M e. W ) -> ( G e. UMGraph -> M =/= N ) ) )
20 19 adantr 
 |-  ( ( X e. dom E /\ Fun ( E |` { X } ) ) -> ( ( ( E ` X ) = { M , N } /\ M e. W ) -> ( G e. UMGraph -> M =/= N ) ) )
21 8 20 mpcom 
 |-  ( ( ( E ` X ) = { M , N } /\ M e. W ) -> ( G e. UMGraph -> M =/= N ) )
22 21 ex 
 |-  ( ( E ` X ) = { M , N } -> ( M e. W -> ( G e. UMGraph -> M =/= N ) ) )
23 22 com13 
 |-  ( G e. UMGraph -> ( M e. W -> ( ( E ` X ) = { M , N } -> M =/= N ) ) )
24 23 imp 
 |-  ( ( G e. UMGraph /\ M e. W ) -> ( ( E ` X ) = { M , N } -> M =/= N ) )