theorem elall (A: set) (l x: nat): $ all A l -> x IN l -> x e. A $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eleq1 |
a1 = x -> (a1 e. lmems l <-> x e. lmems l) |
| 2 |
1 |
conv lmem |
a1 = x -> (a1 e. lmems l <-> x IN l) |
| 3 |
|
eleq1 |
a1 = x -> (a1 e. A <-> x e. A) |
| 4 |
2, 3 |
imeqd |
a1 = x -> (a1 e. lmems l -> a1 e. A <-> x IN l -> x e. A) |
| 5 |
4 |
eale |
A. a1 (a1 e. lmems l -> a1 e. A) -> x IN l -> x e. A |
| 6 |
5 |
conv all, subset |
all A l -> x IN l -> x e. A |
Axiom use
axs_prop_calc
(ax_1,
ax_2,
ax_3,
ax_mp),
axs_pred_calc
(ax_gen,
ax_4,
ax_5,
ax_6,
ax_7,
ax_10,
ax_12),
axs_set
(ax_8)