theorem elrapp (F: set) (a b: nat): $ b e. F @' a <-> a, b e. F $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqidd |
_1 = b -> a = a |
| 2 |
|
id |
_1 = b -> _1 = b |
| 3 |
1, 2 |
preqd |
_1 = b -> a, _1 = a, b |
| 4 |
|
eqsidd |
_1 = b -> F == F |
| 5 |
3, 4 |
eleqd |
_1 = b -> (a, _1 e. F <-> a, b e. F) |
| 6 |
5 |
elabe |
b e. {_1 | a, _1 e. F} <-> a, b e. F |
| 7 |
6 |
conv rapp |
b e. F @' a <-> a, b 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
(peano2,
addeq,
muleq)