theorem cplv: $ Compl _V == 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
binth |
~x e. Compl _V -> ~x e. 0 -> (x e. Compl _V <-> x e. 0) |
| 2 |
|
con2 |
(x e. Compl _V -> ~x e. _V) -> x e. _V -> ~x e. Compl _V |
| 3 |
|
bi1 |
(x e. Compl _V <-> ~x e. _V) -> x e. Compl _V -> ~x e. _V |
| 4 |
|
elcpl |
x e. Compl _V <-> ~x e. _V |
| 5 |
3, 4 |
ax_mp |
x e. Compl _V -> ~x e. _V |
| 6 |
2, 5 |
ax_mp |
x e. _V -> ~x e. Compl _V |
| 7 |
|
elv |
x e. _V |
| 8 |
6, 7 |
ax_mp |
~x e. Compl _V |
| 9 |
1, 8 |
ax_mp |
~x e. 0 -> (x e. Compl _V <-> x e. 0) |
| 10 |
|
el02 |
~x e. 0 |
| 11 |
9, 10 |
ax_mp |
x e. Compl _V <-> x e. 0 |
| 12 |
11 |
eqri |
Compl _V == 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)