theorem lmemsnoc (a b x: nat): $ x IN a |> b <-> x IN a \/ x = b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr |
(x IN a |> b <-> x IN a \/ x IN b : 0) -> (x IN a \/ x IN b : 0 <-> x IN a \/ x = b) -> (x IN a |> b <-> x IN a \/ x = b) |
| 2 |
|
lmemappend |
x IN a ++ b : 0 <-> x IN a \/ x IN b : 0 |
| 3 |
2 |
conv snoc |
x IN a |> b <-> x IN a \/ x IN b : 0 |
| 4 |
1, 3 |
ax_mp |
(x IN a \/ x IN b : 0 <-> x IN a \/ x = b) -> (x IN a |> b <-> x IN a \/ x = b) |
| 5 |
|
lmem1 |
x IN b : 0 <-> x = b |
| 6 |
5 |
oreq2i |
x IN a \/ x IN b : 0 <-> x IN a \/ x = b |
| 7 |
4, 6 |
ax_mp |
x IN a |> b <-> x IN a \/ x = 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)