theorem dmss (A B: set): $ A C_ B -> Dom A C_ Dom B $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
equn1 |
Dom A C_ Dom B <-> Dom A u. Dom B == Dom B |
| 2 |
|
equn1 |
A C_ B <-> A u. B == B |
| 3 |
|
dmun |
Dom (A u. B) == Dom A u. Dom B |
| 4 |
|
dmeq |
A u. B == B -> Dom (A u. B) == Dom B |
| 5 |
3, 4 |
syl5eqsr |
A u. B == B -> Dom A u. Dom B == Dom B |
| 6 |
2, 5 |
sylbi |
A C_ B -> Dom A u. Dom B == Dom B |
| 7 |
1, 6 |
sylibr |
A C_ B -> Dom A C_ Dom 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
(peano2,
addeq,
muleq)