theorem all201 (R: set) (l: nat): $ 0, l e. all2 R <-> l = 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
leneq0 |
len l = 0 <-> l = 0 |
| 2 |
|
len0 |
len 0 = 0 |
| 3 |
|
all2len |
0, l e. all2 R -> len 0 = len l |
| 4 |
3 |
eqcomd |
0, l e. all2 R -> len l = len 0 |
| 5 |
2, 4 |
syl6eq |
0, l e. all2 R -> len l = 0 |
| 6 |
1, 5 |
sylib |
0, l e. all2 R -> l = 0 |
| 7 |
|
all20 |
0, 0 e. all2 R |
| 8 |
|
preq2 |
l = 0 -> 0, l = 0, 0 |
| 9 |
8 |
eleq1d |
l = 0 -> (0, l e. all2 R <-> 0, 0 e. all2 R) |
| 10 |
7, 9 |
mpbiri |
l = 0 -> 0, l e. all2 R |
| 11 |
6, 10 |
ibii |
0, l e. all2 R <-> l = 0 |
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
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)