theorem Listss (A B: set): $ A C_ B -> List A C_ List B $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
elList |
a1 e. List A <-> all A a1 |
| 2 |
|
elList |
a1 e. List B <-> all B a1 |
| 3 |
|
ssall |
A C_ B -> all A a1 -> all B a1 |
| 4 |
2, 3 |
syl6ibr |
A C_ B -> all A a1 -> a1 e. List B |
| 5 |
1, 4 |
syl5bi |
A C_ B -> a1 e. List A -> a1 e. List B |
| 6 |
5 |
iald |
A C_ B -> A. a1 (a1 e. List A -> a1 e. List B) |
| 7 |
6 |
conv subset |
A C_ B -> List A C_ List 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)