theorem leastle (A: set) (a: nat): $ a e. A -> least A <= a $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
leastlem |
a e. A -> least A e. A /\ A. x (x e. A -> least A <= x) |
| 2 |
|
eleq1 |
x = a -> (x e. A <-> a e. A) |
| 3 |
|
leeq2 |
x = a -> (least A <= x <-> least A <= a) |
| 4 |
2, 3 |
imeqd |
x = a -> (x e. A -> least A <= x <-> a e. A -> least A <= a) |
| 5 |
4 |
eale |
A. x (x e. A -> least A <= x) -> a e. A -> least A <= a |
| 6 |
5 |
anwr |
least A e. A /\ A. x (x e. A -> least A <= x) -> a e. A -> least A <= a |
| 7 |
6 |
com12 |
a e. A -> least A e. A /\ A. x (x e. A -> least A <= x) -> least A <= a |
| 8 |
1, 7 |
mpd |
a e. A -> least A <= 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),
axs_peano
(peano1,
peano2,
peano5,
addeq,
add0,
addS)