Detailed syntax breakdown of Definition df-salg
Step | Hyp | Ref
| Expression |
1 | | csalg 43552 |
. 2
class
SAlg |
2 | | c0 4251 |
. . . . 5
class
∅ |
3 | | vx |
. . . . . 6
setvar 𝑥 |
4 | 3 | cv 1542 |
. . . . 5
class 𝑥 |
5 | 2, 4 | wcel 2111 |
. . . 4
wff ∅
∈ 𝑥 |
6 | 4 | cuni 4833 |
. . . . . . 7
class ∪ 𝑥 |
7 | | vy |
. . . . . . . 8
setvar 𝑦 |
8 | 7 | cv 1542 |
. . . . . . 7
class 𝑦 |
9 | 6, 8 | cdif 3877 |
. . . . . 6
class (∪ 𝑥
∖ 𝑦) |
10 | 9, 4 | wcel 2111 |
. . . . 5
wff (∪ 𝑥
∖ 𝑦) ∈ 𝑥 |
11 | 10, 7, 4 | wral 3062 |
. . . 4
wff
∀𝑦 ∈
𝑥 (∪ 𝑥
∖ 𝑦) ∈ 𝑥 |
12 | | com 7662 |
. . . . . . 7
class
ω |
13 | | cdom 8644 |
. . . . . . 7
class
≼ |
14 | 8, 12, 13 | wbr 5067 |
. . . . . 6
wff 𝑦 ≼
ω |
15 | 8 | cuni 4833 |
. . . . . . 7
class ∪ 𝑦 |
16 | 15, 4 | wcel 2111 |
. . . . . 6
wff ∪ 𝑦
∈ 𝑥 |
17 | 14, 16 | wi 4 |
. . . . 5
wff (𝑦 ≼ ω → ∪ 𝑦
∈ 𝑥) |
18 | 4 | cpw 4527 |
. . . . 5
class 𝒫
𝑥 |
19 | 17, 7, 18 | wral 3062 |
. . . 4
wff
∀𝑦 ∈
𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦
∈ 𝑥) |
20 | 5, 11, 19 | w3a 1089 |
. . 3
wff (∅
∈ 𝑥 ∧
∀𝑦 ∈ 𝑥 (∪
𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦
∈ 𝑥)) |
21 | 20, 3 | cab 2715 |
. 2
class {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦
∈ 𝑥))} |
22 | 1, 21 | wceq 1543 |
1
wff SAlg =
{𝑥 ∣ (∅ ∈
𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦
∈ 𝑥))} |