theorem unincpl (A B: set): $ A u. B i^i Compl A == A u. B $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eqstr |
A u. B i^i Compl A == (A u. B) i^i (A u. Compl A) -> (A u. B) i^i (A u. Compl A) == A u. B -> A u. B i^i Compl A == A u. B |
| 2 |
|
undi |
A u. B i^i Compl A == (A u. B) i^i (A u. Compl A) |
| 3 |
1, 2 |
ax_mp |
(A u. B) i^i (A u. Compl A) == A u. B -> A u. B i^i Compl A == A u. B |
| 4 |
|
eqin1 |
A u. B C_ A u. Compl A <-> (A u. B) i^i (A u. Compl A) == A u. B |
| 5 |
|
sseq2 |
A u. Compl A == _V -> (A u. B C_ A u. Compl A <-> A u. B C_ _V) |
| 6 |
|
uncpl2 |
A u. Compl A == _V |
| 7 |
5, 6 |
ax_mp |
A u. B C_ A u. Compl A <-> A u. B C_ _V |
| 8 |
|
ssv2 |
A u. B C_ _V |
| 9 |
7, 8 |
mpbir |
A u. B C_ A u. Compl A |
| 10 |
4, 9 |
mpbi |
(A u. B) i^i (A u. Compl A) == A u. B |
| 11 |
3, 10 |
ax_mp |
A u. B i^i Compl A == A u. 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)