Description: Define "point-finite." (Contributed by Jeff Hankins, 21-Jan-2010)
Ref | Expression | ||
---|---|---|---|
Assertion | df-ptfin | ⊢ PtFin = { 𝑥 ∣ ∀ 𝑦 ∈ ∪ 𝑥 { 𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧 } ∈ Fin } |
Step | Hyp | Ref | Expression |
---|---|---|---|
0 | cptfin | ⊢ PtFin | |
1 | vx | ⊢ 𝑥 | |
2 | vy | ⊢ 𝑦 | |
3 | 1 | cv | ⊢ 𝑥 |
4 | 3 | cuni | ⊢ ∪ 𝑥 |
5 | vz | ⊢ 𝑧 | |
6 | 2 | cv | ⊢ 𝑦 |
7 | 5 | cv | ⊢ 𝑧 |
8 | 6 7 | wcel | ⊢ 𝑦 ∈ 𝑧 |
9 | 8 5 3 | crab | ⊢ { 𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧 } |
10 | cfn | ⊢ Fin | |
11 | 9 10 | wcel | ⊢ { 𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧 } ∈ Fin |
12 | 11 2 4 | wral | ⊢ ∀ 𝑦 ∈ ∪ 𝑥 { 𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧 } ∈ Fin |
13 | 12 1 | cab | ⊢ { 𝑥 ∣ ∀ 𝑦 ∈ ∪ 𝑥 { 𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧 } ∈ Fin } |
14 | 0 13 | wceq | ⊢ PtFin = { 𝑥 ∣ ∀ 𝑦 ∈ ∪ 𝑥 { 𝑧 ∈ 𝑥 ∣ 𝑦 ∈ 𝑧 } ∈ Fin } |