Metamath Proof Explorer

Theorem wwlksnndef

Description: Conditions for WWalksN not being defined. (Contributed by Alexander van der Vekens, 30-Jul-2018) (Revised by AV, 19-Apr-2021)

Ref Expression
Assertion wwlksnndef 
|- ( ( G e/ _V \/ N e/ NN0 ) -> ( N WWalksN G ) = (/) )

Proof

Step Hyp Ref Expression
1 neq0 
 |-  ( -. ( N WWalksN G ) = (/) <-> E. w w e. ( N WWalksN G ) )
2 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
3 2 wwlknbp 
 |-  ( w e. ( N WWalksN G ) -> ( G e. _V /\ N e. NN0 /\ w e. Word ( Vtx ` G ) ) )
4 nnel 
 |-  ( -. G e/ _V <-> G e. _V )
5 nnel 
 |-  ( -. N e/ NN0 <-> N e. NN0 )
6 4 5 anbi12i 
 |-  ( ( -. G e/ _V /\ -. N e/ NN0 ) <-> ( G e. _V /\ N e. NN0 ) )
7 6 biimpri 
 |-  ( ( G e. _V /\ N e. NN0 ) -> ( -. G e/ _V /\ -. N e/ NN0 ) )
8 7 3adant3 
 |-  ( ( G e. _V /\ N e. NN0 /\ w e. Word ( Vtx ` G ) ) -> ( -. G e/ _V /\ -. N e/ NN0 ) )
9 ioran 
 |-  ( -. ( G e/ _V \/ N e/ NN0 ) <-> ( -. G e/ _V /\ -. N e/ NN0 ) )
10 8 9 sylibr 
 |-  ( ( G e. _V /\ N e. NN0 /\ w e. Word ( Vtx ` G ) ) -> -. ( G e/ _V \/ N e/ NN0 ) )
11 3 10 syl 
 |-  ( w e. ( N WWalksN G ) -> -. ( G e/ _V \/ N e/ NN0 ) )
12 11 exlimiv 
 |-  ( E. w w e. ( N WWalksN G ) -> -. ( G e/ _V \/ N e/ NN0 ) )
13 1 12 sylbi 
 |-  ( -. ( N WWalksN G ) = (/) -> -. ( G e/ _V \/ N e/ NN0 ) )
14 13 con4i 
 |-  ( ( G e/ _V \/ N e/ NN0 ) -> ( N WWalksN G ) = (/) )