theorem Arrow02 (A: set): $ ~A == 0 -> Arrow A 0 == 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eq0al |
Arrow A 0 == 0 <-> A. a1 ~a1 e. Arrow A 0 |
| 2 |
|
con3 |
(a1 e. Arrow A 0 -> A == 0) -> ~A == 0 -> ~a1 e. Arrow A 0 |
| 3 |
|
elArrow |
a1 e. Arrow A 0 <-> func a1 A 0 |
| 4 |
|
func03 |
func a1 A 0 <-> a1 == 0 /\ A == 0 |
| 5 |
|
anr |
a1 == 0 /\ A == 0 -> A == 0 |
| 6 |
4, 5 |
sylbi |
func a1 A 0 -> A == 0 |
| 7 |
3, 6 |
sylbi |
a1 e. Arrow A 0 -> A == 0 |
| 8 |
2, 7 |
ax_mp |
~A == 0 -> ~a1 e. Arrow A 0 |
| 9 |
8 |
iald |
~A == 0 -> A. a1 ~a1 e. Arrow A 0 |
| 10 |
1, 9 |
sylibr |
~A == 0 -> Arrow A 0 == 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)