Definition df-aleph 4827

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 4874, alephsuc 4877, and alephlim 4875. 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	|- aleph = rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)

Distinct variable group:   x,y,z

Detailed syntax breakdown of Definition df-aleph

Step	Hyp	Ref	Expression
1		cale 4824	. 2 class aleph
2		vy	. . . . . 6 set y
3	2	cv 957	. . . . 5 class y
4		vx	. . . . . . . . 9 set x
5	4	cv 957	. . . . . . . 8 class x
6		vz	. . . . . . . . 9 set z
7	6	cv 957	. . . . . . . 8 class z
8		csdm 4372	. . . . . . . 8 class ~<
9	5, 7, 8	wbr 2624	. . . . . . 7 wff x ~< z
10		con0 2954	. . . . . . 7 class On
11	9, 6, 10	crab 1651	. . . . . 6 class {z e. On | x ~< z}
12	11	cint 2537	. . . . 5 class |^|{z e. On | x ~< z}
13	3, 12	wceq 958	. . . 4 wff y = |^|{z e. On | x ~< z}
14	13, 4, 2	copab 2671	. . 3 class {<.x, y>. | y = |^|{z e. On | x ~< z}}
15		com 3137	. . 3 class om
16	14, 15	crdg 3937	. 2 class rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)
17	1, 16	wceq 958	1 wff aleph = rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)

This definition is referenced by:  alephfnon 4873  aleph0 4874  alephlim 4875  alephon 4876  alephsuc 4877

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