theorem ltirr (a: nat): $ ~a < a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(0 < 0 <-> 0 + a < 0 + a) -> (0 + a < 0 + a <-> a < a) -> (0 < 0 <-> a < a) |
| 2 |
|
ltadd1 |
0 < 0 <-> 0 + a < 0 + a |
| 3 |
1, 2 |
ax_mp |
(0 + a < 0 + a <-> a < a) -> (0 < 0 <-> a < a) |
| 4 |
|
lteq |
0 + a = a -> 0 + a = a -> (0 + a < 0 + a <-> a < a) |
| 5 |
|
add01 |
0 + a = a |
| 6 |
4, 5 |
ax_mp |
0 + a = a -> (0 + a < 0 + a <-> a < a) |
| 7 |
6, 5 |
ax_mp |
0 + a < 0 + a <-> a < a |
| 8 |
3, 7 |
ax_mp |
0 < 0 <-> a < a |
| 9 |
|
lt02 |
~0 < 0 |
| 10 |
8, 9 |
mtbi |
~a < 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_peano
(peano1,
peano2,
peano5,
addeq,
add0,
addS)