pub theorem srecpval {i: nat} (A: set) (n: nat):
$ srecp A n <-> n, sep (upto n) {i | srecp A i} e. A $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(srecp A n <-> true (nat (n, sep (upto n) {i | srecp A i} e. A))) ->
(true (nat (n, sep (upto n) {i | srecp A i} e. A)) <-> n, sep (upto n) {i | srecp A i} e. A) ->
(srecp A n <-> n, sep (upto n) {i | srecp A i} e. A) |
| 2 |
|
trueeq |
srecpaux A n = nat (n, sep (upto n) {i | srecp A i} e. A) -> (true (srecpaux A n) <-> true (nat (n, sep (upto n) {i | srecp A i} e. A))) |
| 3 |
2 |
conv srecp |
srecpaux A n = nat (n, sep (upto n) {i | srecp A i} e. A) -> (srecp A n <-> true (nat (n, sep (upto n) {i | srecp A i} e. A))) |
| 4 |
|
srecpauxval |
srecpaux A n = nat (n, sep (upto n) {i | srecp A i} e. A) |
| 5 |
3, 4 |
ax_mp |
srecp A n <-> true (nat (n, sep (upto n) {i | srecp A i} e. A)) |
| 6 |
1, 5 |
ax_mp |
(true (nat (n, sep (upto n) {i | srecp A i} e. A)) <-> n, sep (upto n) {i | srecp A i} e. A) -> (srecp A n <-> n, sep (upto n) {i | srecp A i} e. A) |
| 7 |
|
truenat |
true (nat (n, sep (upto n) {i | srecp A i} e. A)) <-> n, sep (upto n) {i | srecp A i} e. A |
| 8 |
6, 7 |
ax_mp |
srecp A n <-> n, sep (upto n) {i | srecp A i} e. 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)