theorem sapp01 (a: nat): $ 0 @@ a == 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eq0al |
0 @@ a == 0 <-> A. x ~x e. 0 @@ a |
| 2 |
|
eleq1 |
fst x, snd x = x -> (fst x, snd x e. 0 @@ a <-> x e. 0 @@ a) |
| 3 |
|
fstsnd |
fst x, snd x = x |
| 4 |
2, 3 |
ax_mp |
fst x, snd x e. 0 @@ a <-> x e. 0 @@ a |
| 5 |
|
prelsapp |
fst x, snd x e. 0 @@ a <-> (a, fst x), snd x e. 0 |
| 6 |
|
el02 |
~(a, fst x), snd x e. 0 |
| 7 |
5, 6 |
mtbir |
~fst x, snd x e. 0 @@ a |
| 8 |
4, 7 |
mtbi |
~x e. 0 @@ a |
| 9 |
8 |
ax_gen |
A. x ~x e. 0 @@ a |
| 10 |
1, 9 |
mpbir |
0 @@ 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)