theorem xpvv: $ Xp _V _V == _V $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bith |
p e. Xp _V _V -> p e. _V -> (p e. Xp _V _V <-> p e. _V) |
| 2 |
|
elxp |
p e. Xp _V _V <-> fst p e. _V /\ snd p e. _V |
| 3 |
|
ian |
fst p e. _V -> snd p e. _V -> fst p e. _V /\ snd p e. _V |
| 4 |
|
elv |
fst p e. _V |
| 5 |
3, 4 |
ax_mp |
snd p e. _V -> fst p e. _V /\ snd p e. _V |
| 6 |
|
elv |
snd p e. _V |
| 7 |
5, 6 |
ax_mp |
fst p e. _V /\ snd p e. _V |
| 8 |
2, 7 |
mpbir |
p e. Xp _V _V |
| 9 |
1, 8 |
ax_mp |
p e. _V -> (p e. Xp _V _V <-> p e. _V) |
| 10 |
|
elv |
p e. _V |
| 11 |
9, 10 |
ax_mp |
p e. Xp _V _V <-> p e. _V |
| 12 |
11 |
eqri |
Xp _V _V == _V |
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)