theorem sninj (a b: nat): $ sn a = sn b <-> a = b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
thesn |
the (sn a) = a |
| 2 |
|
thesn |
the (sn b) = b |
| 3 |
|
nseq |
sn a = sn b -> sn a == sn b |
| 4 |
3 |
theeqd |
sn a = sn b -> the (sn a) = the (sn b) |
| 5 |
1, 2, 4 |
eqtr3g |
sn a = sn b -> a = b |
| 6 |
|
sneq |
a = b -> sn a = sn b |
| 7 |
5, 6 |
ibii |
sn a = sn b <-> 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)