Theorem upgr1wlkdlem2

Description: Lemma 2 for upgr1wlkd . (Contributed by AV, 22-Jan-2021)

	Ref	Expression
Hypotheses	upgr1wlkd.p	|- P = <" X Y ">
	upgr1wlkd.f	|- F = <" J ">
	upgr1wlkd.x	|- ( ph -> X e. ( Vtx ` G ) )
	upgr1wlkd.y	|- ( ph -> Y e. ( Vtx ` G ) )
	upgr1wlkd.j	|- ( ph -> ( ( iEdg ` G ) ` J ) = { X , Y } )
Assertion	upgr1wlkdlem2	|- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( ( iEdg ` G ) ` J ) )

Proof

Step	Hyp	Ref	Expression
1		upgr1wlkd.p	|- P = <" X Y ">
2		upgr1wlkd.f	|- F = <" J ">
3		upgr1wlkd.x	|- ( ph -> X e. ( Vtx ` G ) )
4		upgr1wlkd.y	|- ( ph -> Y e. ( Vtx ` G ) )
5		upgr1wlkd.j	|- ( ph -> ( ( iEdg ` G ) ` J ) = { X , Y } )
6		ssid	|- { X , Y } C_ { X , Y }
7		sseq2	|- ( ( ( iEdg ` G ) ` J ) = { X , Y } -> ( { X , Y } C_ ( ( iEdg ` G ) ` J ) <-> { X , Y } C_ { X , Y } ) )
8	7	adantl	|- ( ( ( ph /\ X =/= Y ) /\ ( ( iEdg ` G ) ` J ) = { X , Y } ) -> ( { X , Y } C_ ( ( iEdg ` G ) ` J ) <-> { X , Y } C_ { X , Y } ) )
9	6 8	mpbiri	|- ( ( ( ph /\ X =/= Y ) /\ ( ( iEdg ` G ) ` J ) = { X , Y } ) -> { X , Y } C_ ( ( iEdg ` G ) ` J ) )
10	5 9	mpidan	|- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( ( iEdg ` G ) ` J ) )