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Mirrors > Home > ILE Home > Th. List > df-nninf | GIF version |
Description: Define the set of nonincreasing sequences in 2o ↑𝑚 ω. Definition in Section 3.1 of [Pierik], p. 15. If we assumed excluded middle, this would be essentially the same as ℕ0* as defined at df-xnn0 9041 but in its absence the relationship between the two is more complicated. This definition would function much the same whether we used ω or ℕ0, but the former allows us to take advantage of 2o = {∅, 1o} (df2o3 6327) so we adopt it. (Contributed by Jim Kingdon, 14-Jul-2022.) |
Ref | Expression |
---|---|
df-nninf | ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xnninf 7005 | . 2 class ℕ∞ | |
2 | vi | . . . . . . . 8 setvar 𝑖 | |
3 | 2 | cv 1330 | . . . . . . 7 class 𝑖 |
4 | 3 | csuc 4287 | . . . . . 6 class suc 𝑖 |
5 | vf | . . . . . . 7 setvar 𝑓 | |
6 | 5 | cv 1330 | . . . . . 6 class 𝑓 |
7 | 4, 6 | cfv 5123 | . . . . 5 class (𝑓‘suc 𝑖) |
8 | 3, 6 | cfv 5123 | . . . . 5 class (𝑓‘𝑖) |
9 | 7, 8 | wss 3071 | . . . 4 wff (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) |
10 | com 4504 | . . . 4 class ω | |
11 | 9, 2, 10 | wral 2416 | . . 3 wff ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) |
12 | c2o 6307 | . . . 4 class 2o | |
13 | cmap 6542 | . . . 4 class ↑𝑚 | |
14 | 12, 10, 13 | co 5774 | . . 3 class (2o ↑𝑚 ω) |
15 | 11, 5, 14 | crab 2420 | . 2 class {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} |
16 | 1, 15 | wceq 1331 | 1 wff ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)} |
Colors of variables: wff set class |
This definition is referenced by: infnninf 7022 nnnninf 7023 0nninf 13197 nninff 13198 nnsf 13199 peano4nninf 13200 nninfalllemn 13202 nninfalllem1 13203 nninfex 13205 nninfself 13209 |
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