theorem lower0 (A: set): $ ~finite A -> lower A = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
absurd |
~a1 == A -> a1 == A -> a1 = 0 |
| 2 |
|
con3 |
(a1 == A -> finite A) -> ~finite A -> ~a1 == A |
| 3 |
|
finns |
finite a1 |
| 4 |
|
fineq |
a1 == A -> (finite a1 <-> finite A) |
| 5 |
3, 4 |
mpbii |
a1 == A -> finite A |
| 6 |
2, 5 |
ax_mp |
~finite A -> ~a1 == A |
| 7 |
1, 6 |
syl |
~finite A -> a1 == A -> a1 = 0 |
| 8 |
7 |
eqthe0abd |
~finite A -> the {a1 | a1 == A} = 0 |
| 9 |
8 |
conv lower |
~finite A -> lower 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,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)