theorem cplinj (A B: set): $ A == B <-> Compl A == Compl B $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
cpleq |
A == B -> Compl A == Compl B |
| 2 |
|
eqseq |
Compl (Compl A) == A -> Compl (Compl B) == B -> (Compl (Compl A) == Compl (Compl B) <-> A == B) |
| 3 |
|
cplcpl |
Compl (Compl A) == A |
| 4 |
2, 3 |
ax_mp |
Compl (Compl B) == B -> (Compl (Compl A) == Compl (Compl B) <-> A == B) |
| 5 |
|
cplcpl |
Compl (Compl B) == B |
| 6 |
4, 5 |
ax_mp |
Compl (Compl A) == Compl (Compl B) <-> A == B |
| 7 |
|
cpleq |
Compl A == Compl B -> Compl (Compl A) == Compl (Compl B) |
| 8 |
6, 7 |
sylib |
Compl A == Compl B -> A == B |
| 9 |
1, 8 |
ibii |
A == B <-> Compl A == 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)