theorem theo0 {x y: nat} (A: set): $ ~E. y A == {x | x = y} <-> theo A = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
theo01 |
~E. y A == {x | x = y} -> theo A = 0 |
| 2 |
|
con2 |
(E. y A == {x | x = y} -> ~theo A = 0) -> theo A = 0 -> ~E. y A == {x | x = y} |
| 3 |
|
sucne0 |
theo A = suc y -> theo A != 0 |
| 4 |
3 |
conv ne |
theo A = suc y -> ~theo A = 0 |
| 5 |
|
theoid1 |
A == {x | x = y} -> theo A = suc y |
| 6 |
4, 5 |
syl |
A == {x | x = y} -> ~theo A = 0 |
| 7 |
6 |
eex |
E. y A == {x | x = y} -> ~theo A = 0 |
| 8 |
2, 7 |
ax_mp |
theo A = 0 -> ~E. y A == {x | x = y} |
| 9 |
1, 8 |
ibii |
~E. y A == {x | x = y} <-> theo A = 0 |
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)