Step |
Hyp |
Ref |
Expression |
1 |
|
ackbij.f |
|- F = ( x e. ( ~P _om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) ) |
2 |
1
|
ackbij1lem10 |
|- F : ( ~P _om i^i Fin ) --> _om |
3 |
|
peano1 |
|- (/) e. _om |
4 |
2 3
|
f0cli |
|- ( F ` (/) ) e. _om |
5 |
|
nna0 |
|- ( ( F ` (/) ) e. _om -> ( ( F ` (/) ) +o (/) ) = ( F ` (/) ) ) |
6 |
4 5
|
ax-mp |
|- ( ( F ` (/) ) +o (/) ) = ( F ` (/) ) |
7 |
|
un0 |
|- ( (/) u. (/) ) = (/) |
8 |
7
|
fveq2i |
|- ( F ` ( (/) u. (/) ) ) = ( F ` (/) ) |
9 |
|
ackbij1lem3 |
|- ( (/) e. _om -> (/) e. ( ~P _om i^i Fin ) ) |
10 |
3 9
|
ax-mp |
|- (/) e. ( ~P _om i^i Fin ) |
11 |
|
in0 |
|- ( (/) i^i (/) ) = (/) |
12 |
1
|
ackbij1lem9 |
|- ( ( (/) e. ( ~P _om i^i Fin ) /\ (/) e. ( ~P _om i^i Fin ) /\ ( (/) i^i (/) ) = (/) ) -> ( F ` ( (/) u. (/) ) ) = ( ( F ` (/) ) +o ( F ` (/) ) ) ) |
13 |
10 10 11 12
|
mp3an |
|- ( F ` ( (/) u. (/) ) ) = ( ( F ` (/) ) +o ( F ` (/) ) ) |
14 |
6 8 13
|
3eqtr2ri |
|- ( ( F ` (/) ) +o ( F ` (/) ) ) = ( ( F ` (/) ) +o (/) ) |
15 |
|
nnacan |
|- ( ( ( F ` (/) ) e. _om /\ ( F ` (/) ) e. _om /\ (/) e. _om ) -> ( ( ( F ` (/) ) +o ( F ` (/) ) ) = ( ( F ` (/) ) +o (/) ) <-> ( F ` (/) ) = (/) ) ) |
16 |
4 4 3 15
|
mp3an |
|- ( ( ( F ` (/) ) +o ( F ` (/) ) ) = ( ( F ` (/) ) +o (/) ) <-> ( F ` (/) ) = (/) ) |
17 |
14 16
|
mpbi |
|- ( F ` (/) ) = (/) |