Definition df-nninf 7097

Description: Define the set of nonincreasing sequences in 2_o ↑_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 9199 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 2_o = {∅, 1_o} (df2o3 6409) so we adopt it. (Contributed by Jim Kingdon, 14-Jul-2022.)

Assertion

Ref	Expression
df-nninf	⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)}

Distinct variable group:   𝑓,𝑖

Detailed syntax breakdown of Definition df-nninf

Step	Hyp	Ref	Expression
1		xnninf 7096	. 2 class ℕ∞
2		vi	. . . . . . . 8 setvar 𝑖
3	2	cv 1347	. . . . . . 7 class 𝑖
4	3	csuc 4350	. . . . . 6 class suc 𝑖
5		vf	. . . . . . 7 setvar 𝑓
6	5	cv 1347	. . . . . 6 class 𝑓
7	4, 6	cfv 5198	. . . . 5 class (𝑓‘suc 𝑖)
8	3, 6	cfv 5198	. . . . 5 class (𝑓‘𝑖)
9	7, 8	wss 3121	. . . 4 wff (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)
10		com 4574	. . . 4 class ω
11	9, 2, 10	wral 2448	. . 3 wff ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)
12		c2o 6389	. . . 4 class 2o
13		cmap 6626	. . . 4 class ↑𝑚
14	12, 10, 13	co 5853	. . 3 class (2o ↑𝑚 ω)
15	11, 5, 14	crab 2452	. 2 class {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)}
16	1, 15	wceq 1348	1 wff ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)}

This definition is referenced by:  nninfex  7098  nninff  7099  infnninf  7100  infnninfOLD  7101  nnnninf  7102  nnnninfeq  7104  nnnninfeq2  7105  nninfwlpoimlemg  7151  0nninf  14037  nnsf  14038  peano4nninf  14039  nninfalllem1  14041  nninfself  14046