theorem least0 (A: set) {x: nat}: $ ~E. x x e. A -> least A = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
absurdr |
E. x x e. A -> ~E. x x e. A -> y = 0 |
| 2 |
|
eleq1 |
x = y -> (x e. A <-> y e. A) |
| 3 |
2 |
iexe |
y e. A -> E. x x e. A |
| 4 |
3 |
anwl |
y e. A /\ A. z (z e. A -> y <= z) -> E. x x e. A |
| 5 |
1, 4 |
syl |
y e. A /\ A. z (z e. A -> y <= z) -> ~E. x x e. A -> y = 0 |
| 6 |
5 |
com12 |
~E. x x e. A -> y e. A /\ A. z (z e. A -> y <= z) -> y = 0 |
| 7 |
6 |
eqthe0abd |
~E. x x e. A -> the {y | y e. A /\ A. z (z e. A -> y <= z)} = 0 |
| 8 |
7 |
conv least |
~E. x x e. A -> least 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)