Definition df-spac 6275

Description: Define the special class generator for the disproof of the axiom of choice. Definition 6.1 of [Specker] p. 973. (Contributed by SF, 3-Mar-2015.)

Assertion

Ref	Expression
df-spac	|- Spac = (m e. NC |-> Clos1 ({m}, {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))}))

Distinct variable group:   x,m,y

Detailed syntax breakdown of Definition df-spac

Step	Hyp	Ref	Expression
1		cspac 6274	. 2 class Spac
2		vm	. . 3 setvar m
3		cncs 6089	. . 3 class NC
4	2	cv 1641	. . . . 5 class m
5	4	csn 3738	. . . 4 class {m}
6		vx	. . . . . . . 8 setvar x
7	6	cv 1641	. . . . . . 7 class x
8	7, 3	wcel 1710	. . . . . 6 wff x e. NC
9		vy	. . . . . . . 8 setvar y
10	9	cv 1641	. . . . . . 7 class y
11	10, 3	wcel 1710	. . . . . 6 wff y e. NC
12		c2c 6095	. . . . . . . 8 class 2c
13		cce 6097	. . . . . . . 8 class ↑c
14	12, 7, 13	co 5526	. . . . . . 7 class (2c ↑c x)
15	10, 14	wceq 1642	. . . . . 6 wff y = (2c ↑c x)
16	8, 11, 15	w3a 934	. . . . 5 wff (x e. NC /\ y e. NC /\ y = (2c ↑c x))
17	16, 6, 9	copab 4623	. . . 4 class {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))}
18	5, 17	cclos1 5873	. . 3 class Clos1 ({m}, {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))})
19	2, 3, 18	cmpt 5652	. 2 class (m e. NC |-> Clos1 ({m}, {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))}))
20	1, 19	wceq 1642	1 wff Spac = (m e. NC |-> Clos1 ({m}, {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))}))

This definition is referenced by:  spacval  6283  fnspac  6284