pub theorem elsep (n: nat) (A: set) (a: nat):
$ a e. sep n A <-> a e. n /\ a e. A $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(a e. sep n A <-> a e. n i^i A) -> (a e. n i^i A <-> a e. n /\ a e. A) -> (a e. sep n A <-> a e. n /\ a e. A) |
| 2 |
|
ellower |
finite (n i^i A) -> (a e. lower (n i^i A) <-> a e. n i^i A) |
| 3 |
2 |
conv sep |
finite (n i^i A) -> (a e. sep n A <-> a e. n i^i A) |
| 4 |
|
finss |
n i^i A C_ n -> finite n -> finite (n i^i A) |
| 5 |
|
inss1 |
n i^i A C_ n |
| 6 |
4, 5 |
ax_mp |
finite n -> finite (n i^i A) |
| 7 |
|
finns |
finite n |
| 8 |
6, 7 |
ax_mp |
finite (n i^i A) |
| 9 |
3, 8 |
ax_mp |
a e. sep n A <-> a e. n i^i A |
| 10 |
1, 9 |
ax_mp |
(a e. n i^i A <-> a e. n /\ a e. A) -> (a e. sep n A <-> a e. n /\ a e. A) |
| 11 |
|
elin |
a e. n i^i A <-> a e. n /\ a e. A |
| 12 |
10, 11 |
ax_mp |
a e. sep n A <-> a e. n /\ a 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)