Definition df-cv 11851

Description: Define the covers relation (on the Hilbert lattice). Definition 3.2.18 of [PtakPulmannova] p. 68, whose notation we use. Ptak/Pulmannova's notation A <o B is read "B covers A " or "A is covered by B " , and it means that B is larger than A and there is nothing in between. See cvbr 11854 and cvbr2 11855 for membership relations.

Assertion

Ref	Expression
df-cv	|- <o = {<.x, y>. | ((x e. CH /\ y e. CH) /\ (x C. y /\ -. E.z e. CH (x C. z /\ z C. y)))}

Distinct variable group:   x,y,z

Detailed syntax breakdown of Definition df-cv

Step	Hyp	Ref	Expression
1		ccv 10466	. 2 class <o
2		vx	. . . . . . 7 set x
3	2	cv 1297	. . . . . 6 class x
4		cch 10430	. . . . . 6 class CH
5	3, 4	wcel 1300	. . . . 5 wff x e. CH
6		vy	. . . . . . 7 set y
7	6	cv 1297	. . . . . 6 class y
8	7, 4	wcel 1300	. . . . 5 wff y e. CH
9	5, 8	wa 240	. . . 4 wff (x e. CH /\ y e. CH)
10	3, 7	wpss 2594	. . . . 5 wff x C. y
11		vz	. . . . . . . . . 10 set z
12	11	cv 1297	. . . . . . . . 9 class z
13	3, 12	wpss 2594	. . . . . . . 8 wff x C. z
14	12, 7	wpss 2594	. . . . . . . 8 wff z C. y
15	13, 14	wa 240	. . . . . . 7 wff (x C. z /\ z C. y)
16	15, 11, 4	wrex 2106	. . . . . 6 wff E.z e. CH (x C. z /\ z C. y)
17	16	wn 2	. . . . 5 wff -. E.z e. CH (x C. z /\ z C. y)
18	10, 17	wa 240	. . . 4 wff (x C. y /\ -. E.z e. CH (x C. z /\ z C. y))
19	9, 18	wa 240	. . 3 wff ((x e. CH /\ y e. CH) /\ (x C. y /\ -. E.z e. CH (x C. z /\ z C. y)))
20	19, 2, 6	copab 3395	. 2 class {<.x, y>. | ((x e. CH /\ y e. CH) /\ (x C. y /\ -. E.z e. CH (x C. z /\ z C. y)))}
21	1, 20	wceq 1298	1 wff <o = {<.x, y>. | ((x e. CH /\ y e. CH) /\ (x C. y /\ -. E.z e. CH (x C. z /\ z C. y)))}

This definition is referenced by:  cvbr 11854

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