Theorem 2wlkdlem7

Description: Lemma 7 for 2wlkd . (Contributed by AV, 14-Feb-2021)

	Ref	Expression
Hypotheses	2wlkd.p	|- P = <" A B C ">
	2wlkd.f	|- F = <" J K ">
	2wlkd.s	|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
	2wlkd.n	|- ( ph -> ( A =/= B /\ B =/= C ) )
	2wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
Assertion	2wlkdlem7	|- ( ph -> ( J e. _V /\ K e. _V ) )

Proof

Step	Hyp	Ref	Expression
1		2wlkd.p	|- P = <" A B C ">
2		2wlkd.f	|- F = <" J K ">
3		2wlkd.s	|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
4		2wlkd.n	|- ( ph -> ( A =/= B /\ B =/= C ) )
5		2wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
6	1 2 3 4 5	2wlkdlem6	|- ( ph -> ( B e. ( I ` J ) /\ B e. ( I ` K ) ) )
7		elfvex	|- ( B e. ( I ` J ) -> J e. _V )
8		elfvex	|- ( B e. ( I ` K ) -> K e. _V )
9	7 8	anim12i	|- ( ( B e. ( I ` J ) /\ B e. ( I ` K ) ) -> ( J e. _V /\ K e. _V ) )
10	6 9	syl	|- ( ph -> ( J e. _V /\ K e. _V ) )