theorem xppi221 (A B C D: set) (a: nat):
$ a e. Xp A (Xp B (Xp C D)) -> pi221 a e. C $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
xppi22 |
a e. Xp A (Xp B (Xp C D)) -> pi22 a e. Xp C D |
| 2 |
|
xpfst |
pi22 a e. Xp C D -> fst (pi22 a) e. C |
| 3 |
2 |
conv pi221 |
pi22 a e. Xp C D -> pi221 a e. C |
| 4 |
1, 3 |
rsyl |
a e. Xp A (Xp B (Xp C D)) -> pi221 a e. C |
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)