theorem cplun (A B: set): $ Compl (A u. B) == Compl A i^i Compl B $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqstr |
Compl (A u. B) == Compl (Compl (Compl A i^i Compl B)) -> Compl (Compl (Compl A i^i Compl B)) == Compl A i^i Compl B -> Compl (A u. B) == Compl A i^i Compl B |
| 2 |
|
cpleq |
A u. B == Compl (Compl A i^i Compl B) -> Compl (A u. B) == Compl (Compl (Compl A i^i Compl B)) |
| 3 |
|
eqstr2 |
Compl (Compl A i^i Compl B) == Compl (Compl A) u. Compl (Compl B) -> Compl (Compl A) u. Compl (Compl B) == A u. B -> A u. B == Compl (Compl A i^i Compl B) |
| 4 |
|
cplin |
Compl (Compl A i^i Compl B) == Compl (Compl A) u. Compl (Compl B) |
| 5 |
3, 4 |
ax_mp |
Compl (Compl A) u. Compl (Compl B) == A u. B -> A u. B == Compl (Compl A i^i Compl B) |
| 6 |
|
uneq |
Compl (Compl A) == A -> Compl (Compl B) == B -> Compl (Compl A) u. Compl (Compl B) == A u. B |
| 7 |
|
cplcpl |
Compl (Compl A) == A |
| 8 |
6, 7 |
ax_mp |
Compl (Compl B) == B -> Compl (Compl A) u. Compl (Compl B) == A u. B |
| 9 |
|
cplcpl |
Compl (Compl B) == B |
| 10 |
8, 9 |
ax_mp |
Compl (Compl A) u. Compl (Compl B) == A u. B |
| 11 |
5, 10 |
ax_mp |
A u. B == Compl (Compl A i^i Compl B) |
| 12 |
2, 11 |
ax_mp |
Compl (A u. B) == Compl (Compl (Compl A i^i Compl B)) |
| 13 |
1, 12 |
ax_mp |
Compl (Compl (Compl A i^i Compl B)) == Compl A i^i Compl B -> Compl (A u. B) == Compl A i^i Compl B |
| 14 |
|
cplcpl |
Compl (Compl (Compl A i^i Compl B)) == Compl A i^i Compl B |
| 15 |
13, 14 |
ax_mp |
Compl (A u. B) == Compl A i^i Compl B |
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)