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Metamath Proof Explorer

Theorem usgrnloopALT

Description: Alternate proof of usgrnloop , not using umgrnloop . (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) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypothesis usgrnloopv.e 
|- E = ( iEdg ` G )
Assertion usgrnloopALT 
|- ( 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 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
3 1 2 usgredgprv 
 |-  ( ( G e. USGraph /\ x e. dom E ) -> ( ( E ` x ) = { M , N } -> ( M e. ( Vtx ` G ) /\ N e. ( Vtx ` G ) ) ) )
4 3 imp 
 |-  ( ( ( G e. USGraph /\ x e. dom E ) /\ ( E ` x ) = { M , N } ) -> ( M e. ( Vtx ` G ) /\ N e. ( Vtx ` G ) ) )
5 1 usgrnloopv 
 |-  ( ( G e. USGraph /\ M e. ( Vtx ` G ) ) -> ( ( E ` x ) = { M , N } -> M =/= N ) )
6 5 ex 
 |-  ( G e. USGraph -> ( M e. ( Vtx ` G ) -> ( ( E ` x ) = { M , N } -> M =/= N ) ) )
7 6 com23 
 |-  ( G e. USGraph -> ( ( E ` x ) = { M , N } -> ( M e. ( Vtx ` G ) -> M =/= N ) ) )
8 7 adantr 
 |-  ( ( G e. USGraph /\ x e. dom E ) -> ( ( E ` x ) = { M , N } -> ( M e. ( Vtx ` G ) -> M =/= N ) ) )
9 8 imp 
 |-  ( ( ( G e. USGraph /\ x e. dom E ) /\ ( E ` x ) = { M , N } ) -> ( M e. ( Vtx ` G ) -> M =/= N ) )
10 9 com12 
 |-  ( M e. ( Vtx ` G ) -> ( ( ( G e. USGraph /\ x e. dom E ) /\ ( E ` x ) = { M , N } ) -> M =/= N ) )
11 10 adantr 
 |-  ( ( M e. ( Vtx ` G ) /\ N e. ( Vtx ` G ) ) -> ( ( ( G e. USGraph /\ x e. dom E ) /\ ( E ` x ) = { M , N } ) -> M =/= N ) )
12 4 11 mpcom 
 |-  ( ( ( G e. USGraph /\ x e. dom E ) /\ ( E ` x ) = { M , N } ) -> M =/= N )
13 12 ex 
 |-  ( ( G e. USGraph /\ x e. dom E ) -> ( ( E ` x ) = { M , N } -> M =/= N ) )
14 13 rexlimdva 
 |-  ( G e. USGraph -> ( E. x e. dom E ( E ` x ) = { M , N } -> M =/= N ) )