theorem elArray1 (A: set) (a: nat): $ a : 0 e. Array A 1 <-> a e. A $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(a : 0 e. Array A 1 <-> a e. A /\ 0 e. Array A 0) -> (a e. A /\ 0 e. Array A 0 <-> a e. A) -> (a : 0 e. Array A 1 <-> a e. A) |
| 2 |
|
elArrayS |
a : 0 e. Array A (suc 0) <-> a e. A /\ 0 e. Array A 0 |
| 3 |
2 |
conv d1 |
a : 0 e. Array A 1 <-> a e. A /\ 0 e. Array A 0 |
| 4 |
1, 3 |
ax_mp |
(a e. A /\ 0 e. Array A 0 <-> a e. A) -> (a : 0 e. Array A 1 <-> a e. A) |
| 5 |
|
bian2 |
0 e. Array A 0 -> (a e. A /\ 0 e. Array A 0 <-> a e. A) |
| 6 |
|
elArray0 |
0 e. Array A 0 |
| 7 |
5, 6 |
ax_mp |
a e. A /\ 0 e. Array A 0 <-> a e. A |
| 8 |
4, 7 |
ax_mp |
a : 0 e. Array A 1 <-> a e. A |
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)