theorem elArraySne0 (A: set) (l n: nat): $ l e. Array A (suc n) -> l != 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
elArray |
l e. Array A (suc n) <-> l e. List A /\ len l = suc n |
| 2 |
|
noteq |
(len l = 0 <-> l = 0) -> (~len l = 0 <-> ~l = 0) |
| 3 |
2 |
conv ne |
(len l = 0 <-> l = 0) -> (~len l = 0 <-> l != 0) |
| 4 |
|
leneq0 |
len l = 0 <-> l = 0 |
| 5 |
3, 4 |
ax_mp |
~len l = 0 <-> l != 0 |
| 6 |
|
sucne0 |
len l = suc n -> len l != 0 |
| 7 |
6 |
conv ne |
len l = suc n -> ~len l = 0 |
| 8 |
5, 7 |
sylib |
len l = suc n -> l != 0 |
| 9 |
8 |
anwr |
l e. List A /\ len l = suc n -> l != 0 |
| 10 |
1, 9 |
sylbi |
l e. Array A (suc n) -> l != 0 |
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)