theorem natinj (p q: wff): $ p <-> q <-> nat p = nat q $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr4 |
(p <-> q <-> nat p <= nat q /\ nat q <= nat p) -> (nat p = nat q <-> nat p <= nat q /\ nat q <= nat p) -> (p <-> q <-> nat p = nat q) |
| 2 |
|
aneq |
(p -> q <-> nat p <= nat q) -> (q -> p <-> nat q <= nat p) -> ((p -> q) /\ (q -> p) <-> nat p <= nat q /\ nat q <= nat p) |
| 3 |
2 |
conv iff |
(p -> q <-> nat p <= nat q) -> (q -> p <-> nat q <= nat p) -> (p <-> q <-> nat p <= nat q /\ nat q <= nat p) |
| 4 |
|
natle |
p -> q <-> nat p <= nat q |
| 5 |
3, 4 |
ax_mp |
(q -> p <-> nat q <= nat p) -> (p <-> q <-> nat p <= nat q /\ nat q <= nat p) |
| 6 |
|
natle |
q -> p <-> nat q <= nat p |
| 7 |
5, 6 |
ax_mp |
p <-> q <-> nat p <= nat q /\ nat q <= nat p |
| 8 |
1, 7 |
ax_mp |
(nat p = nat q <-> nat p <= nat q /\ nat q <= nat p) -> (p <-> q <-> nat p = nat q) |
| 9 |
|
eqlele |
nat p = nat q <-> nat p <= nat q /\ nat q <= nat p |
| 10 |
8, 9 |
ax_mp |
p <-> q <-> nat p = nat q |
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),
axs_peano
(peano1,
peano2,
peano5,
addeq,
add0,
addS)