theorem writefin (F: set) (a b: nat): $ finite F -> finite (write F a b) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
fineq |
write F a b == (F |` {x | x != a}) u. sn (a, b) -> (finite (write F a b) <-> finite ((F |` {x | x != a}) u. sn (a, b))) |
| 2 |
|
writeres |
write F a b == (F |` {x | x != a}) u. sn (a, b) |
| 3 |
1, 2 |
ax_mp |
finite (write F a b) <-> finite ((F |` {x | x != a}) u. sn (a, b)) |
| 4 |
|
unfin |
finite (F |` {x | x != a}) -> finite (sn (a, b)) -> finite ((F |` {x | x != a}) u. sn (a, b)) |
| 5 |
|
resfin |
finite F -> finite (F |` {x | x != a}) |
| 6 |
|
finns |
finite (sn (a, b)) |
| 7 |
6 |
a1i |
finite F -> finite (sn (a, b)) |
| 8 |
4, 5, 7 |
sylc |
finite F -> finite ((F |` {x | x != a}) u. sn (a, b)) |
| 9 |
3, 8 |
sylibr |
finite F -> finite (write F 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)