theorem equn1 (A B: set): $ A C_ B <-> A u. B == B $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
unss |
A u. B C_ B <-> A C_ B /\ B C_ B |
| 2 |
|
id |
A C_ B -> A C_ B |
| 3 |
|
ssid |
B C_ B |
| 4 |
3 |
a1i |
A C_ B -> B C_ B |
| 5 |
2, 4 |
iand |
A C_ B -> A C_ B /\ B C_ B |
| 6 |
1, 5 |
sylibr |
A C_ B -> A u. B C_ B |
| 7 |
|
ssun2 |
B C_ A u. B |
| 8 |
7 |
a1i |
A C_ B -> B C_ A u. B |
| 9 |
6, 8 |
ssasymd |
A C_ B -> A u. B == B |
| 10 |
|
ssun1 |
A C_ A u. B |
| 11 |
|
sseq2 |
A u. B == B -> (A C_ A u. B <-> A C_ B) |
| 12 |
10, 11 |
mpbii |
A u. B == B -> A C_ B |
| 13 |
9, 12 |
ibii |
A C_ B <-> A u. B == 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)