theorem ss02 (A: set): $ A C_ 0 <-> A == 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
ss01 |
0 C_ A |
| 2 |
|
ssasym |
A C_ 0 -> 0 C_ A -> A == 0 |
| 3 |
1, 2 |
mpi |
A C_ 0 -> A == 0 |
| 4 |
|
ss01 |
0 C_ 0 |
| 5 |
|
sseq1 |
A == 0 -> (A C_ 0 <-> 0 C_ 0) |
| 6 |
4, 5 |
mpbiri |
A == 0 -> A C_ 0 |
| 7 |
3, 6 |
ibii |
A C_ 0 <-> A == 0 |
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)