Step |
Hyp |
Ref |
Expression |
1 |
|
brdomi |
|- ( _om ~<_ A -> E. f f : _om -1-1-> A ) |
2 |
|
peano1 |
|- (/) e. _om |
3 |
|
f1f1orn |
|- ( f : _om -1-1-> A -> f : _om -1-1-onto-> ran f ) |
4 |
3
|
adantr |
|- ( ( f : _om -1-1-> A /\ A = (/) ) -> f : _om -1-1-onto-> ran f ) |
5 |
|
f1f |
|- ( f : _om -1-1-> A -> f : _om --> A ) |
6 |
5
|
frnd |
|- ( f : _om -1-1-> A -> ran f C_ A ) |
7 |
|
sseq0 |
|- ( ( ran f C_ A /\ A = (/) ) -> ran f = (/) ) |
8 |
6 7
|
sylan |
|- ( ( f : _om -1-1-> A /\ A = (/) ) -> ran f = (/) ) |
9 |
8
|
f1oeq3d |
|- ( ( f : _om -1-1-> A /\ A = (/) ) -> ( f : _om -1-1-onto-> ran f <-> f : _om -1-1-onto-> (/) ) ) |
10 |
4 9
|
mpbid |
|- ( ( f : _om -1-1-> A /\ A = (/) ) -> f : _om -1-1-onto-> (/) ) |
11 |
|
f1ocnv |
|- ( f : _om -1-1-onto-> (/) -> `' f : (/) -1-1-onto-> _om ) |
12 |
|
noel |
|- -. (/) e. (/) |
13 |
|
f1o00 |
|- ( `' f : (/) -1-1-onto-> _om <-> ( `' f = (/) /\ _om = (/) ) ) |
14 |
13
|
simprbi |
|- ( `' f : (/) -1-1-onto-> _om -> _om = (/) ) |
15 |
14
|
eleq2d |
|- ( `' f : (/) -1-1-onto-> _om -> ( (/) e. _om <-> (/) e. (/) ) ) |
16 |
12 15
|
mtbiri |
|- ( `' f : (/) -1-1-onto-> _om -> -. (/) e. _om ) |
17 |
10 11 16
|
3syl |
|- ( ( f : _om -1-1-> A /\ A = (/) ) -> -. (/) e. _om ) |
18 |
2 17
|
mt2 |
|- -. ( f : _om -1-1-> A /\ A = (/) ) |
19 |
18
|
imnani |
|- ( f : _om -1-1-> A -> -. A = (/) ) |
20 |
19
|
neqned |
|- ( f : _om -1-1-> A -> A =/= (/) ) |
21 |
20
|
exlimiv |
|- ( E. f f : _om -1-1-> A -> A =/= (/) ) |
22 |
1 21
|
syl |
|- ( _om ~<_ A -> A =/= (/) ) |