theorem elpower (a b: nat): $ a e. power b <-> a C_ b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(a e. power b <-> a e. Power b) -> (a e. Power b <-> a C_ b) -> (a e. power b <-> a C_ b) |
| 2 |
|
ellower |
finite (Power b) -> (a e. lower (Power b) <-> a e. Power b) |
| 3 |
2 |
conv power |
finite (Power b) -> (a e. power b <-> a e. Power b) |
| 4 |
|
powerfin |
finite b -> finite (Power b) |
| 5 |
|
finns |
finite b |
| 6 |
4, 5 |
ax_mp |
finite (Power b) |
| 7 |
3, 6 |
ax_mp |
a e. power b <-> a e. Power b |
| 8 |
1, 7 |
ax_mp |
(a e. Power b <-> a C_ b) -> (a e. power b <-> a C_ b) |
| 9 |
|
elPower |
a e. Power b <-> a C_ b |
| 10 |
8, 9 |
ax_mp |
a e. power b <-> a C_ 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_set
(elab,
ax_8),
axs_the
(theid,
the0),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)