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Metamath Proof Explorer

Theorem clwwlkneq0

Description: Sufficient conditions for ClWWalksN to be empty. (Contributed by Alexander van der Vekens, 15-Sep-2018) (Revised by AV, 24-Apr-2021) (Proof shortened by AV, 24-Feb-2022)

Ref Expression
Assertion clwwlkneq0 
|- ( ( G e/ _V \/ N e/ NN ) -> ( N ClWWalksN G ) = (/) )

Proof

Step Hyp Ref Expression
1 df-nel 
 |-  ( G e/ _V <-> -. G e. _V )
2 ianor 
 |-  ( -. ( N e. NN0 /\ N =/= 0 ) <-> ( -. N e. NN0 \/ -. N =/= 0 ) )
3 1 2 orbi12i 
 |-  ( ( G e/ _V \/ -. ( N e. NN0 /\ N =/= 0 ) ) <-> ( -. G e. _V \/ ( -. N e. NN0 \/ -. N =/= 0 ) ) )
4 df-nel 
 |-  ( N e/ NN <-> -. N e. NN )
5 elnnne0 
 |-  ( N e. NN <-> ( N e. NN0 /\ N =/= 0 ) )
6 4 5 xchbinx 
 |-  ( N e/ NN <-> -. ( N e. NN0 /\ N =/= 0 ) )
7 6 orbi2i 
 |-  ( ( G e/ _V \/ N e/ NN ) <-> ( G e/ _V \/ -. ( N e. NN0 /\ N =/= 0 ) ) )
8 orass 
 |-  ( ( ( -. G e. _V \/ -. N e. NN0 ) \/ -. N =/= 0 ) <-> ( -. G e. _V \/ ( -. N e. NN0 \/ -. N =/= 0 ) ) )
9 3 7 8 3bitr4i 
 |-  ( ( G e/ _V \/ N e/ NN ) <-> ( ( -. G e. _V \/ -. N e. NN0 ) \/ -. N =/= 0 ) )
10 ianor 
 |-  ( -. ( N e. NN0 /\ G e. _V ) <-> ( -. N e. NN0 \/ -. G e. _V ) )
11 orcom 
 |-  ( ( -. N e. NN0 \/ -. G e. _V ) <-> ( -. G e. _V \/ -. N e. NN0 ) )
12 10 11 bitri 
 |-  ( -. ( N e. NN0 /\ G e. _V ) <-> ( -. G e. _V \/ -. N e. NN0 ) )
13 df-clwwlkn 
 |-  ClWWalksN = ( n e. NN0 , g e. _V |-> { w e. ( ClWWalks ` g ) | ( # ` w ) = n } )
14 13 mpondm0 
 |-  ( -. ( N e. NN0 /\ G e. _V ) -> ( N ClWWalksN G ) = (/) )
15 12 14 sylbir 
 |-  ( ( -. G e. _V \/ -. N e. NN0 ) -> ( N ClWWalksN G ) = (/) )
16 nne 
 |-  ( -. N =/= 0 <-> N = 0 )
17 oveq1 
 |-  ( N = 0 -> ( N ClWWalksN G ) = ( 0 ClWWalksN G ) )
18 clwwlkn0 
 |-  ( 0 ClWWalksN G ) = (/)
19 17 18 eqtrdi 
 |-  ( N = 0 -> ( N ClWWalksN G ) = (/) )
20 16 19 sylbi 
 |-  ( -. N =/= 0 -> ( N ClWWalksN G ) = (/) )
21 15 20 jaoi 
 |-  ( ( ( -. G e. _V \/ -. N e. NN0 ) \/ -. N =/= 0 ) -> ( N ClWWalksN G ) = (/) )
22 9 21 sylbi 
 |-  ( ( G e/ _V \/ N e/ NN ) -> ( N ClWWalksN G ) = (/) )