theorem el12 (a: nat): $ a e. 1 <-> a = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
lt12 |
a < 1 <-> a = 0 |
| 2 |
|
ellt |
a e. 1 -> a < 1 |
| 3 |
1, 2 |
sylib |
a e. 1 -> a = 0 |
| 4 |
|
el01 |
0 e. 1 <-> odd 1 |
| 5 |
|
odd1 |
odd 1 |
| 6 |
4, 5 |
mpbir |
0 e. 1 |
| 7 |
|
eleq1 |
a = 0 -> (a e. 1 <-> 0 e. 1) |
| 8 |
6, 7 |
mpbiri |
a = 0 -> a e. 1 |
| 9 |
3, 8 |
ibii |
a e. 1 <-> a = 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)