theorem ineq0 (A B: set): $ A i^i B == 0 <-> A C_ Compl B $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr3 |
(A i^i Compl (Compl B) == 0 <-> A i^i B == 0) -> (A i^i Compl (Compl B) == 0 <-> A C_ Compl B) -> (A i^i B == 0 <-> A C_ Compl B) |
| 2 |
|
eqseq1 |
A i^i Compl (Compl B) == A i^i B -> (A i^i Compl (Compl B) == 0 <-> A i^i B == 0) |
| 3 |
|
ineq2 |
Compl (Compl B) == B -> A i^i Compl (Compl B) == A i^i B |
| 4 |
|
cplcpl |
Compl (Compl B) == B |
| 5 |
3, 4 |
ax_mp |
A i^i Compl (Compl B) == A i^i B |
| 6 |
2, 5 |
ax_mp |
A i^i Compl (Compl B) == 0 <-> A i^i B == 0 |
| 7 |
1, 6 |
ax_mp |
(A i^i Compl (Compl B) == 0 <-> A C_ Compl B) -> (A i^i B == 0 <-> A C_ Compl B) |
| 8 |
|
incpleq0 |
A i^i Compl (Compl B) == 0 <-> A C_ Compl B |
| 9 |
7, 8 |
ax_mp |
A i^i B == 0 <-> A C_ 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),
axs_the
(theid,
the0),
axs_peano
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)