theorem ltlene (a b: nat): $ a < b <-> a <= b /\ a != b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
ltle |
a < b -> a <= b |
| 2 |
|
ltne |
a < b -> a != b |
| 3 |
1, 2 |
iand |
a < b -> a <= b /\ a != b |
| 4 |
|
bi1 |
(a <= b <-> ~a < b -> a = b) -> a <= b -> ~a < b -> a = b |
| 5 |
|
leloe |
a <= b <-> a < b \/ a = b |
| 6 |
5 |
conv or |
a <= b <-> ~a < b -> a = b |
| 7 |
4, 6 |
ax_mp |
a <= b -> ~a < b -> a = b |
| 8 |
7 |
con1d |
a <= b -> ~a = b -> a < b |
| 9 |
8 |
conv ne |
a <= b -> a != b -> a < b |
| 10 |
9 |
imp |
a <= b /\ a != b -> a < b |
| 11 |
3, 10 |
ibii |
a < b <-> a <= b /\ a != 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
(peano1,
peano2,
peano5,
addeq,
add0,
addS)