theorem ltsuc (a b: nat): $ a < b <-> suc a < suc b $;
Step | Hyp | Ref | Expression |
1 |
|
bitr |
(a < b <-> a + 1 < b + 1) -> (a + 1 < b + 1 <-> suc a < suc b) -> (a < b <-> suc a < suc b) |
2 |
|
ltadd1 |
a < b <-> a + 1 < b + 1 |
3 |
1, 2 |
ax_mp |
(a + 1 < b + 1 <-> suc a < suc b) -> (a < b <-> suc a < suc b) |
4 |
|
lteq |
a + 1 = suc a -> b + 1 = suc b -> (a + 1 < b + 1 <-> suc a < suc b) |
5 |
|
add12 |
a + 1 = suc a |
6 |
4, 5 |
ax_mp |
b + 1 = suc b -> (a + 1 < b + 1 <-> suc a < suc b) |
7 |
|
add12 |
b + 1 = suc b |
8 |
6, 7 |
ax_mp |
a + 1 < b + 1 <-> suc a < suc b |
9 |
3, 8 |
ax_mp |
a < b <-> suc a < suc b |
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
(peano2,
peano5,
addeq,
add0,
addS)