theorem allS (A: set) (a b: nat): $ all A (a : b) <-> a e. A /\ all A b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(all A (a : b) <-> a ; lmems b C_ A) -> (a ; lmems b C_ A <-> a e. A /\ all A b) -> (all A (a : b) <-> a e. A /\ all A b) |
| 2 |
|
sseq1 |
lmems (a : b) == a ; lmems b -> (lmems (a : b) C_ A <-> a ; lmems b C_ A) |
| 3 |
2 |
conv all |
lmems (a : b) == a ; lmems b -> (all A (a : b) <-> a ; lmems b C_ A) |
| 4 |
|
nseq |
lmems (a : b) = a ; lmems b -> lmems (a : b) == a ; lmems b |
| 5 |
|
lmemsS |
lmems (a : b) = a ; lmems b |
| 6 |
4, 5 |
ax_mp |
lmems (a : b) == a ; lmems b |
| 7 |
3, 6 |
ax_mp |
all A (a : b) <-> a ; lmems b C_ A |
| 8 |
1, 7 |
ax_mp |
(a ; lmems b C_ A <-> a e. A /\ all A b) -> (all A (a : b) <-> a e. A /\ all A b) |
| 9 |
|
insss |
a ; lmems b C_ A <-> a e. A /\ lmems b C_ A |
| 10 |
9 |
conv all |
a ; lmems b C_ A <-> a e. A /\ all A b |
| 11 |
8, 10 |
ax_mp |
all A (a : b) <-> a e. A /\ all A 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
(peano1,
peano2,
peano5,
addeq,
muleq,
add0,
addS,
mul0,
mulS)