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Mirrors > Home > MPE Home > Th. List > df-fin1a | Structured version Visualization version GIF version |
Description: A set is Ia-finite iff it is not the union of two I-infinite sets. Equivalent to definition Ia of [Levy58] p. 2. A I-infinite Ia-finite set is also known as an amorphous set. This is the second of Levy's eight definitions of finite set. Levy's I-finite is equivalent to our df-fin 8725 and not repeated here. These eight definitions are equivalent with Choice but strictly decreasing in strength in models where Choice fails; conversely, they provide a series of increasingly stronger notions of infiniteness. (Contributed by Stefan O'Rear, 12-Nov-2014.) |
Ref | Expression |
---|---|
df-fin1a | ⊢ FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfin1a 10022 | . 2 class FinIa | |
2 | vy | . . . . . . 7 setvar 𝑦 | |
3 | 2 | cv 1538 | . . . . . 6 class 𝑦 |
4 | cfn 8721 | . . . . . 6 class Fin | |
5 | 3, 4 | wcel 2106 | . . . . 5 wff 𝑦 ∈ Fin |
6 | vx | . . . . . . . 8 setvar 𝑥 | |
7 | 6 | cv 1538 | . . . . . . 7 class 𝑥 |
8 | 7, 3 | cdif 3884 | . . . . . 6 class (𝑥 ∖ 𝑦) |
9 | 8, 4 | wcel 2106 | . . . . 5 wff (𝑥 ∖ 𝑦) ∈ Fin |
10 | 5, 9 | wo 844 | . . . 4 wff (𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin) |
11 | 7 | cpw 4534 | . . . 4 class 𝒫 𝑥 |
12 | 10, 2, 11 | wral 3064 | . . 3 wff ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin) |
13 | 12, 6 | cab 2715 | . 2 class {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)} |
14 | 1, 13 | wceq 1539 | 1 wff FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)} |
Colors of variables: wff setvar class |
This definition is referenced by: isfin1a 10036 |
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