| Step |
Hyp |
Ref |
Expression |
| 1 |
|
1wlkd.p |
|- P = <" X Y "> |
| 2 |
|
1wlkd.f |
|- F = <" J "> |
| 3 |
|
1wlkd.x |
|- ( ph -> X e. V ) |
| 4 |
|
1wlkd.y |
|- ( ph -> Y e. V ) |
| 5 |
|
1wlkd.l |
|- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } ) |
| 6 |
|
1wlkd.j |
|- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) ) |
| 7 |
|
snidg |
|- ( X e. V -> X e. { X } ) |
| 8 |
3 7
|
syl |
|- ( ph -> X e. { X } ) |
| 9 |
8
|
adantr |
|- ( ( ph /\ X = Y ) -> X e. { X } ) |
| 10 |
9 5
|
eleqtrrd |
|- ( ( ph /\ X = Y ) -> X e. ( I ` J ) ) |
| 11 |
4
|
adantr |
|- ( ( ph /\ X =/= Y ) -> Y e. V ) |
| 12 |
|
prssg |
|- ( ( X e. V /\ Y e. V ) -> ( ( X e. ( I ` J ) /\ Y e. ( I ` J ) ) <-> { X , Y } C_ ( I ` J ) ) ) |
| 13 |
3 11 12
|
syl2an2r |
|- ( ( ph /\ X =/= Y ) -> ( ( X e. ( I ` J ) /\ Y e. ( I ` J ) ) <-> { X , Y } C_ ( I ` J ) ) ) |
| 14 |
6 13
|
mpbird |
|- ( ( ph /\ X =/= Y ) -> ( X e. ( I ` J ) /\ Y e. ( I ` J ) ) ) |
| 15 |
14
|
simpld |
|- ( ( ph /\ X =/= Y ) -> X e. ( I ` J ) ) |
| 16 |
10 15
|
pm2.61dane |
|- ( ph -> X e. ( I ` J ) ) |