theorem eluni (A: set) (a: nat) {x: nat}:
$ a e. sUnion A <-> E. x (a e. x /\ x e. A) $;
| Step | Hyp | Ref | Expression |
|---|
| 1 |
|
eleq1 |
y = a -> (y e. x <-> a e. x) |
| 2 |
1 |
aneq1d |
y = a -> (y e. x /\ x e. A <-> a e. x /\ x e. A) |
| 3 |
2 |
exeqd |
y = a -> (E. x (y e. x /\ x e. A) <-> E. x (a e. x /\ x e. A)) |
| 4 |
3 |
elabe |
a e. {y | E. x (y e. x /\ x e. A)} <-> E. x (a e. x /\ x e. A) |
| 5 |
4 |
conv sUnion |
a e. sUnion A <-> E. x (a e. x /\ x e. A) |
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)