theorem ocasepS (S: set) (n: nat) (z: wff): $ suc n e. ocasep z S <-> n e. S $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eleq1 |
suc n - 1 = n -> (suc n - 1 e. S <-> n e. S) |
| 2 |
|
sucsub1 |
suc n - 1 = n |
| 3 |
1, 2 |
ax_mp |
suc n - 1 e. S <-> n e. S |
| 4 |
|
ifpneg |
~suc n = 0 -> (ifp (suc n = 0) z (suc n - 1 e. S) <-> suc n - 1 e. S) |
| 5 |
|
peano1 |
suc n != 0 |
| 6 |
5 |
conv ne |
~suc n = 0 |
| 7 |
4, 6 |
ax_mp |
ifp (suc n = 0) z (suc n - 1 e. S) <-> suc n - 1 e. S |
| 8 |
|
eqeq1 |
a1 = suc n -> (a1 = 0 <-> suc n = 0) |
| 9 |
|
biidd |
a1 = suc n -> (z <-> z) |
| 10 |
|
subeq1 |
a1 = suc n -> a1 - 1 = suc n - 1 |
| 11 |
10 |
eleq1d |
a1 = suc n -> (a1 - 1 e. S <-> suc n - 1 e. S) |
| 12 |
8, 9, 11 |
ifpeqd |
a1 = suc n -> (ifp (a1 = 0) z (a1 - 1 e. S) <-> ifp (suc n = 0) z (suc n - 1 e. S)) |
| 13 |
7, 12 |
syl6bb |
a1 = suc n -> (ifp (a1 = 0) z (a1 - 1 e. S) <-> suc n - 1 e. S) |
| 14 |
3, 13 |
syl6bb |
a1 = suc n -> (ifp (a1 = 0) z (a1 - 1 e. S) <-> n e. S) |
| 15 |
14 |
elabe |
suc n e. {a1 | ifp (a1 = 0) z (a1 - 1 e. S)} <-> n e. S |
| 16 |
15 |
conv ocasep |
suc n e. ocasep z S <-> n e. S |
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,
add0,
addS)