theorem snocT (A: set) (a b: nat):
$ a |> b e. List A <-> a e. List A /\ b e. A $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(a |> b e. List A <-> a e. List A /\ b : 0 e. List A) ->
(a e. List A /\ b : 0 e. List A <-> a e. List A /\ b e. A) ->
(a |> b e. List A <-> a e. List A /\ b e. A) |
| 2 |
|
appendT |
a ++ b : 0 e. List A <-> a e. List A /\ b : 0 e. List A |
| 3 |
2 |
conv snoc |
a |> b e. List A <-> a e. List A /\ b : 0 e. List A |
| 4 |
1, 3 |
ax_mp |
(a e. List A /\ b : 0 e. List A <-> a e. List A /\ b e. A) -> (a |> b e. List A <-> a e. List A /\ b e. A) |
| 5 |
|
elList1 |
b : 0 e. List A <-> b e. A |
| 6 |
5 |
aneq2i |
a e. List A /\ b : 0 e. List A <-> a e. List A /\ b e. A |
| 7 |
4, 6 |
ax_mp |
a |> b e. List A <-> a e. List A /\ b e. A |
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)