theorem subsnsn (a: nat): $ subsn (sn a) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
subsneq |
sn a == {a1 | a1 = a} -> (subsn (sn a) <-> subsn {a1 | a1 = a}) |
| 2 |
|
elsn |
a1 e. sn a <-> a1 = a |
| 3 |
2 |
eqab2i |
sn a == {a1 | a1 = a} |
| 4 |
1, 3 |
ax_mp |
subsn (sn a) <-> subsn {a1 | a1 = a} |
| 5 |
|
subsnsn2 |
subsn {a1 | a1 = a} |
| 6 |
4, 5 |
mpbir |
subsn (sn 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_set
(elab,
ax_8),
axs_the
(theid,
the0),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)