theorem incpl2 (A: set): $ A i^i Compl A == 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
cplinj |
A i^i Compl A == 0 <-> Compl (A i^i Compl A) == Compl 0 |
| 2 |
|
eqstr |
Compl (A i^i Compl A) == Compl A u. Compl (Compl A) -> Compl A u. Compl (Compl A) == Compl 0 -> Compl (A i^i Compl A) == Compl 0 |
| 3 |
|
cplin |
Compl (A i^i Compl A) == Compl A u. Compl (Compl A) |
| 4 |
2, 3 |
ax_mp |
Compl A u. Compl (Compl A) == Compl 0 -> Compl (A i^i Compl A) == Compl 0 |
| 5 |
|
eqstr4 |
Compl A u. Compl (Compl A) == _V -> Compl 0 == _V -> Compl A u. Compl (Compl A) == Compl 0 |
| 6 |
|
uncpl2 |
Compl A u. Compl (Compl A) == _V |
| 7 |
5, 6 |
ax_mp |
Compl 0 == _V -> Compl A u. Compl (Compl A) == Compl 0 |
| 8 |
|
cpl0 |
Compl 0 == _V |
| 9 |
7, 8 |
ax_mp |
Compl A u. Compl (Compl A) == Compl 0 |
| 10 |
4, 9 |
ax_mp |
Compl (A i^i Compl A) == Compl 0 |
| 11 |
1, 10 |
mpbir |
A i^i Compl 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)