theorem elb0 (a b: nat): $ a e. b0 b <-> 0 < a /\ a - 1 e. b $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
bitr3 |
(a e. shl b 1 <-> a e. b0 b) -> (a e. shl b 1 <-> 0 < a /\ a - 1 e. b) -> (a e. b0 b <-> 0 < a /\ a - 1 e. b) |
| 2 |
|
elneq2 |
shl b 1 = b0 b -> (a e. shl b 1 <-> a e. b0 b) |
| 3 |
|
shl12 |
shl b 1 = b0 b |
| 4 |
2, 3 |
ax_mp |
a e. shl b 1 <-> a e. b0 b |
| 5 |
1, 4 |
ax_mp |
(a e. shl b 1 <-> 0 < a /\ a - 1 e. b) -> (a e. b0 b <-> 0 < a /\ a - 1 e. b) |
| 6 |
|
elshl |
a e. shl b 1 <-> 1 <= a /\ a - 1 e. b |
| 7 |
6 |
conv d1, lt |
a e. shl b 1 <-> 0 < a /\ a - 1 e. b |
| 8 |
5, 7 |
ax_mp |
a e. b0 b <-> 0 < a /\ a - 1 e. 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)