Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
|- ( A = (/) -> ( rank ` A ) = ( rank ` (/) ) ) |
2 |
|
r1funlim |
|- ( Fun R1 /\ Lim dom R1 ) |
3 |
2
|
simpri |
|- Lim dom R1 |
4 |
|
limomss |
|- ( Lim dom R1 -> _om C_ dom R1 ) |
5 |
3 4
|
ax-mp |
|- _om C_ dom R1 |
6 |
|
peano1 |
|- (/) e. _om |
7 |
5 6
|
sselii |
|- (/) e. dom R1 |
8 |
|
rankonid |
|- ( (/) e. dom R1 <-> ( rank ` (/) ) = (/) ) |
9 |
7 8
|
mpbi |
|- ( rank ` (/) ) = (/) |
10 |
1 9
|
eqtrdi |
|- ( A = (/) -> ( rank ` A ) = (/) ) |
11 |
|
eqimss |
|- ( ( rank ` A ) = (/) -> ( rank ` A ) C_ (/) ) |
12 |
11
|
adantl |
|- ( ( A e. U. ( R1 " On ) /\ ( rank ` A ) = (/) ) -> ( rank ` A ) C_ (/) ) |
13 |
|
simpl |
|- ( ( A e. U. ( R1 " On ) /\ ( rank ` A ) = (/) ) -> A e. U. ( R1 " On ) ) |
14 |
|
rankr1bg |
|- ( ( A e. U. ( R1 " On ) /\ (/) e. dom R1 ) -> ( A C_ ( R1 ` (/) ) <-> ( rank ` A ) C_ (/) ) ) |
15 |
13 7 14
|
sylancl |
|- ( ( A e. U. ( R1 " On ) /\ ( rank ` A ) = (/) ) -> ( A C_ ( R1 ` (/) ) <-> ( rank ` A ) C_ (/) ) ) |
16 |
12 15
|
mpbird |
|- ( ( A e. U. ( R1 " On ) /\ ( rank ` A ) = (/) ) -> A C_ ( R1 ` (/) ) ) |
17 |
|
r10 |
|- ( R1 ` (/) ) = (/) |
18 |
16 17
|
sseqtrdi |
|- ( ( A e. U. ( R1 " On ) /\ ( rank ` A ) = (/) ) -> A C_ (/) ) |
19 |
|
ss0 |
|- ( A C_ (/) -> A = (/) ) |
20 |
18 19
|
syl |
|- ( ( A e. U. ( R1 " On ) /\ ( rank ` A ) = (/) ) -> A = (/) ) |
21 |
20
|
ex |
|- ( A e. U. ( R1 " On ) -> ( ( rank ` A ) = (/) -> A = (/) ) ) |
22 |
10 21
|
impbid2 |
|- ( A e. U. ( R1 " On ) -> ( A = (/) <-> ( rank ` A ) = (/) ) ) |