Metamath Proof Explorer

Theorem 3wlkdlem7

Description: Lemma 7 for 3wlkd . (Contributed by AV, 7-Feb-2021)

Ref Expression
Hypotheses 3wlkd.p 
|- P = <" A B C D ">
3wlkd.f 
|- F = <" J K L ">
3wlkd.s 
|- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
3wlkd.n 
|- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
3wlkd.e 
|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )
Assertion 3wlkdlem7 
|- ( ph -> ( J e. _V /\ K e. _V /\ L e. _V ) )

Proof

Step Hyp Ref Expression
1 3wlkd.p 
 |-  P = <" A B C D ">
2 3wlkd.f 
 |-  F = <" J K L ">
3 3wlkd.s 
 |-  ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
4 3wlkd.n 
 |-  ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
5 3wlkd.e 
 |-  ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )
6 1 2 3 4 5 3wlkdlem6 
 |-  ( ph -> ( A e. ( I ` J ) /\ B e. ( I ` K ) /\ C e. ( I ` L ) ) )
7 elfvex 
 |-  ( A e. ( I ` J ) -> J e. _V )
8 elfvex 
 |-  ( B e. ( I ` K ) -> K e. _V )
9 elfvex 
 |-  ( C e. ( I ` L ) -> L e. _V )
10 7 8 9 3anim123i 
 |-  ( ( A e. ( I ` J ) /\ B e. ( I ` K ) /\ C e. ( I ` L ) ) -> ( J e. _V /\ K e. _V /\ L e. _V ) )
11 6 10 syl 
 |-  ( ph -> ( J e. _V /\ K e. _V /\ L e. _V ) )