Definition df-aleph 3624

Description: Define the aleph function. Our definition expresses Definition 12 of [Suppes] p. 229 in a closed form, from which we derive the recursive definition as theorems aleph0 3669, alephsuc 3672, and alephlim 3670. The aleph function provides a one-to-one, onto mapping from the ordinal numbers to the infinite cardinal numbers. Roughly, any aleph is the smallest infinite cardinal number whose size is strictly greater than any aleph before it.

Assertion

Ref	Expression
df-aleph	⊢ ℵ = rec({⟨x, y⟩∣y = ∩{z ∈ On∣x ≺ z}}, ω)

Distinct variable group(s):   x,y,z

Detailed syntax breakdown of Definition df-aleph

Step	Hyp	Ref	Expression
1		cale 3621	. 2 class ℵ
2		vy	. . . . . 6 set y
3	2	cv 1089	. . . . 5 class y
4		vx	. . . . . . . . 9 set x
5	4	cv 1089	. . . . . . . 8 class x
6		vz	. . . . . . . . 9 set z
7	6	cv 1089	. . . . . . . 8 class z
8		csdm 3273	. . . . . . . 8 class ≺
9	5, 7, 8	wbr 2054	. . . . . . 7 wff x ≺ z
10		con0 2199	. . . . . . 7 class On
11	9, 6, 10	crab 1204	. . . . . 6 class {z ∈ On∣x ≺ z}
12	11	cint 1965	. . . . 5 class ∩{z ∈ On∣x ≺ z}
13	3, 12	wceq 1091	. . . 4 wff y = ∩{z ∈ On∣x ≺ z}
14	13, 4, 2	copab 2055	. . 3 class {⟨x, y⟩∣y = ∩{z ∈ On∣x ≺ z}}
15		com 2372	. . 3 class ω
16	14, 15	crdg 2969	. 2 class rec({⟨x, y⟩∣y = ∩{z ∈ On∣x ≺ z}}, ω)
17	1, 16	wceq 1091	1 wff ℵ = rec({⟨x, y⟩∣y = ∩{z ∈ On∣x ≺ z}}, ω)

This definition is referenced by:  alephfnon 3668  aleph0 3669  alephlim 3670  alephon 3671  alephsuc 3672

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