Theorem upgr1wlkdlem2 27006

Description: Lemma 2 for upgr1wlkd 27007. (Contributed by AV, 22-Jan-2021.)

Hypotheses

Ref	Expression
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

Ref	Expression
upgr1wlkdlem2	|- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( (iEdg ` G ) ` J ) )

Proof of Theorem upgr1wlkdlem2

Step	Hyp	Ref	Expression
1		upgr1wlkd.j	. 2 |- ( ph -> ( (iEdg ` G ) ` J ) = { X , Y } )
2		ssid 3624	. . 3 |- { X , Y } C_ { X , Y }
3		sseq2 3627	. . . 4 |- ( ( (iEdg ` G ) ` J ) = { X , Y } -> ( { X , Y } C_ ( (iEdg ` G ) ` J ) <-> { X , Y } C_ { X , Y } ) )
4	3	adantl 482	. . 3 |- ( ( ( ph /\ X =/= Y ) /\ ( (iEdg ` G ) ` J ) = { X , Y } ) -> ( { X , Y } C_ ( (iEdg ` G ) ` J ) <-> { X , Y } C_ { X , Y } ) )
5	2, 4	mpbiri 248	. 2 |- ( ( ( ph /\ X =/= Y ) /\ ( (iEdg ` G ) ` J ) = { X , Y } ) -> { X , Y } C_ ( (iEdg ` G ) ` J ) )
6	1, 5	mpidan 704	1 |- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( (iEdg ` G ) ` J ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   {cpr 4179   ` cfv 5888   <"cs1 13294   <"cs2 13586  Vtxcvtx 25874  iEdgciedg 25875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588

This theorem is referenced by:  upgr1wlkd  27007  upgr1trld  27008  upgr1pthd  27009  upgr1pthond  27010