theorem prelsapp (F: set) (a b y: nat): $ b, y e. F @@ a <-> (a, b), y e. F $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr3 |
(y e. F @@ a @' b <-> b, y e. F @@ a) -> (y e. F @@ a @' b <-> (a, b), y e. F) -> (b, y e. F @@ a <-> (a, b), y e. F) |
| 2 |
|
elrapp |
y e. F @@ a @' b <-> b, y e. F @@ a |
| 3 |
1, 2 |
ax_mp |
(y e. F @@ a @' b <-> (a, b), y e. F) -> (b, y e. F @@ a <-> (a, b), y e. F) |
| 4 |
|
bitr |
(y e. F @@ a @' b <-> y e. F @' (a, b)) -> (y e. F @' (a, b) <-> (a, b), y e. F) -> (y e. F @@ a @' b <-> (a, b), y e. F) |
| 5 |
|
eleq2 |
F @@ a @' b == F @' (a, b) -> (y e. F @@ a @' b <-> y e. F @' (a, b)) |
| 6 |
|
sapprapp |
F @@ a @' b == F @' (a, b) |
| 7 |
5, 6 |
ax_mp |
y e. F @@ a @' b <-> y e. F @' (a, b) |
| 8 |
4, 7 |
ax_mp |
(y e. F @' (a, b) <-> (a, b), y e. F) -> (y e. F @@ a @' b <-> (a, b), y e. F) |
| 9 |
|
elrapp |
y e. F @' (a, b) <-> (a, b), y e. F |
| 10 |
8, 9 |
ax_mp |
y e. F @@ a @' b <-> (a, b), y e. F |
| 11 |
3, 10 |
ax_mp |
b, y e. F @@ a <-> (a, b), y e. F |
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)