theorem Arrowfin (A B: set): $ finite A -> finite B -> finite (Arrow A B) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
finss |
Arrow A B C_ Power (Xp A B) -> finite (Power (Xp A B)) -> finite (Arrow A B) |
| 2 |
|
Arrowssxp |
Arrow A B C_ Power (Xp A B) |
| 3 |
1, 2 |
ax_mp |
finite (Power (Xp A B)) -> finite (Arrow A B) |
| 4 |
|
powerfin |
finite (Xp A B) -> finite (Power (Xp A B)) |
| 5 |
|
xpfin |
finite A -> finite B -> finite (Xp A B) |
| 6 |
5 |
imp |
finite A /\ finite B -> finite (Xp A B) |
| 7 |
4, 6 |
syl |
finite A /\ finite B -> finite (Power (Xp A B)) |
| 8 |
3, 7 |
syl |
finite A /\ finite B -> finite (Arrow A B) |
| 9 |
8 |
exp |
finite A -> finite B -> finite (Arrow A B) |
Axiom use
axs_prop_calc
(ax_1,
ax_2,
ax_3,
ax_mp,
itru),
axs_pred_calc
(ax_gen,
ax_4,
ax_5,
ax_6,
ax_7,
ax_10,
ax_11,
ax_12),
axs_set
(elab,
ax_8),
axs_the
(theid,
the0),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)