Theorem 1wlkdlem2 26998

Description: Lemma 2 for 1wlkd 27001. (Contributed by AV, 22-Jan-2021.)

Hypotheses

Ref	Expression
1wlkd.p	|- P = <" X Y ">
1wlkd.f	|- F = <" J ">
1wlkd.x	|- ( ph -> X e. V )
1wlkd.y	|- ( ph -> Y e. V )
1wlkd.l	|- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )
1wlkd.j	|- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )

Assertion

Ref	Expression
1wlkdlem2	|- ( ph -> X e. ( I ` J ) )

Proof of Theorem 1wlkdlem2

Step	Hyp	Ref	Expression
1		1wlkd.x	. . . . 5 |- ( ph -> X e. V )
2		snidg 4206	. . . . 5 |- ( X e. V -> X e. { X } )
3	1, 2	syl 17	. . . 4 |- ( ph -> X e. { X } )
4	3	adantr 481	. . 3 |- ( ( ph /\ X = Y ) -> X e. { X } )
5		1wlkd.l	. . 3 |- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )
6	4, 5	eleqtrrd 2704	. 2 |- ( ( ph /\ X = Y ) -> X e. ( I ` J ) )
7		1wlkd.j	. . . 4 |- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )
8		1wlkd.y	. . . . . 6 |- ( ph -> Y e. V )
9	8	adantr 481	. . . . 5 |- ( ( ph /\ X =/= Y ) -> Y e. V )
10		prssg 4350	. . . . 5 |- ( ( X e. V /\ Y e. V ) -> ( ( X e. ( I ` J ) /\ Y e. ( I ` J ) ) <-> { X , Y } C_ ( I ` J ) ) )
11	1, 9, 10	syl2an2r 876	. . . 4 |- ( ( ph /\ X =/= Y ) -> ( ( X e. ( I ` J ) /\ Y e. ( I ` J ) ) <-> { X , Y } C_ ( I ` J ) ) )
12	7, 11	mpbird 247	. . 3 |- ( ( ph /\ X =/= Y ) -> ( X e. ( I ` J ) /\ Y e. ( I ` J ) ) )
13	12	simpld 475	. 2 |- ( ( ph /\ X =/= Y ) -> X e. ( I ` J ) )
14	6, 13	pm2.61dane 2881	1 |- ( ph -> X e. ( I ` J ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888   <"cs1 13294   <"cs2 13586

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-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-pr 4180

This theorem is referenced by:  1wlkdlem3  26999  1wlkdlem4  27000