theorem dmdisj1 (A B: set): $ Dom A i^i Dom B == 0 -> A i^i B == 0 $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
ss02 |
Dom A i^i Dom B C_ 0 <-> Dom A i^i Dom B == 0 |
| 2 |
|
ss02 |
A i^i B C_ 0 <-> A i^i B == 0 |
| 3 |
|
elin |
x, y e. A i^i B <-> x, y e. A /\ x, y e. B |
| 4 |
|
anim |
(x, y e. A -> x e. Dom A) -> (x, y e. B -> x e. Dom B) -> x, y e. A /\ x, y e. B -> x e. Dom A /\ x e. Dom B |
| 5 |
|
preldm |
x, y e. A -> x e. Dom A |
| 6 |
4, 5 |
ax_mp |
(x, y e. B -> x e. Dom B) -> x, y e. A /\ x, y e. B -> x e. Dom A /\ x e. Dom B |
| 7 |
|
preldm |
x, y e. B -> x e. Dom B |
| 8 |
6, 7 |
ax_mp |
x, y e. A /\ x, y e. B -> x e. Dom A /\ x e. Dom B |
| 9 |
|
elin |
x e. Dom A i^i Dom B <-> x e. Dom A /\ x e. Dom B |
| 10 |
|
absurd |
~x e. 0 -> x e. 0 -> x, y e. 0 |
| 11 |
|
el02 |
~x e. 0 |
| 12 |
10, 11 |
ax_mp |
x e. 0 -> x, y e. 0 |
| 13 |
|
ssel |
Dom A i^i Dom B C_ 0 -> x e. Dom A i^i Dom B -> x e. 0 |
| 14 |
12, 13 |
syl6 |
Dom A i^i Dom B C_ 0 -> x e. Dom A i^i Dom B -> x, y e. 0 |
| 15 |
9, 14 |
syl5bir |
Dom A i^i Dom B C_ 0 -> x e. Dom A /\ x e. Dom B -> x, y e. 0 |
| 16 |
8, 15 |
syl5 |
Dom A i^i Dom B C_ 0 -> x, y e. A /\ x, y e. B -> x, y e. 0 |
| 17 |
3, 16 |
syl5bi |
Dom A i^i Dom B C_ 0 -> x, y e. A i^i B -> x, y e. 0 |
| 18 |
17 |
ssrd2 |
Dom A i^i Dom B C_ 0 -> A i^i B C_ 0 |
| 19 |
2, 18 |
sylib |
Dom A i^i Dom B C_ 0 -> A i^i B == 0 |
| 20 |
1, 19 |
sylbir |
Dom A i^i Dom B == 0 -> A i^i B == 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)