theorem nthlmem (a l n: nat): $ nth n l = suc a -> a IN l $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
lmemnth |
a IN l <-> E. a1 nth a1 l = suc a |
| 2 |
|
ntheq1 |
a1 = n -> nth a1 l = nth n l |
| 3 |
2 |
eqeq1d |
a1 = n -> (nth a1 l = suc a <-> nth n l = suc a) |
| 4 |
3 |
iexe |
nth n l = suc a -> E. a1 nth a1 l = suc a |
| 5 |
1, 4 |
sylibr |
nth n l = suc a -> a IN l |
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)