theorem equn2 (A B: set): $ A C_ B <-> B u. A == B $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(A C_ B <-> A u. B == B) -> (A u. B == B <-> B u. A == B) -> (A C_ B <-> B u. A == B) |
| 2 |
|
equn1 |
A C_ B <-> A u. B == B |
| 3 |
1, 2 |
ax_mp |
(A u. B == B <-> B u. A == B) -> (A C_ B <-> B u. A == B) |
| 4 |
|
eqseq1 |
A u. B == B u. A -> (A u. B == B <-> B u. A == B) |
| 5 |
|
uncom |
A u. B == B u. A |
| 6 |
4, 5 |
ax_mp |
A u. B == B <-> B u. A == B |
| 7 |
3, 6 |
ax_mp |
A C_ B <-> B u. A == 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)