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Definition df-np 9150
Description: Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other. A positive real is defined as the lower class of a Dedekind cut. Definition 9-3.1 of [Gleason] p. 121. (Note: This is a "temporary" definition used in the construction of complex numbers df-c 9288, and is intended to be used only by the construction.) (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
Assertion
Ref Expression
df-np  |-  P.  =  { x  |  (
( (/)  C.  x  /\  x  C.  Q. )  /\  A. y  e.  x  ( A. z ( z 
<Q  y  ->  z  e.  x )  /\  E. z  e.  x  y  <Q  z ) ) }
Distinct variable group:    x, y, z

Detailed syntax breakdown of Definition df-np
StepHypRef Expression
1 cnp 9026 . 2  class  P.
2 c0 3637 . . . . . 6  class  (/)
3 vx . . . . . . 7  setvar  x
43cv 1368 . . . . . 6  class  x
52, 4wpss 3329 . . . . 5  wff  (/)  C.  x
6 cnq 9019 . . . . . 6  class  Q.
74, 6wpss 3329 . . . . 5  wff  x  C.  Q.
85, 7wa 369 . . . 4  wff  ( (/)  C.  x  /\  x  C.  Q. )
9 vz . . . . . . . . . 10  setvar  z
109cv 1368 . . . . . . . . 9  class  z
11 vy . . . . . . . . . 10  setvar  y
1211cv 1368 . . . . . . . . 9  class  y
13 cltq 9025 . . . . . . . . 9  class  <Q
1410, 12, 13wbr 4292 . . . . . . . 8  wff  z  <Q 
y
159, 3wel 1757 . . . . . . . 8  wff  z  e.  x
1614, 15wi 4 . . . . . . 7  wff  ( z 
<Q  y  ->  z  e.  x )
1716, 9wal 1367 . . . . . 6  wff  A. z
( z  <Q  y  ->  z  e.  x )
1812, 10, 13wbr 4292 . . . . . . 7  wff  y  <Q 
z
1918, 9, 4wrex 2716 . . . . . 6  wff  E. z  e.  x  y  <Q  z
2017, 19wa 369 . . . . 5  wff  ( A. z ( z  <Q 
y  ->  z  e.  x )  /\  E. z  e.  x  y  <Q  z )
2120, 11, 4wral 2715 . . . 4  wff  A. y  e.  x  ( A. z ( z  <Q 
y  ->  z  e.  x )  /\  E. z  e.  x  y  <Q  z )
228, 21wa 369 . . 3  wff  ( (
(/)  C.  x  /\  x  C. 
Q. )  /\  A. y  e.  x  ( A. z ( z  <Q 
y  ->  z  e.  x )  /\  E. z  e.  x  y  <Q  z ) )
2322, 3cab 2429 . 2  class  { x  |  ( ( (/)  C.  x  /\  x  C.  Q. )  /\  A. y  e.  x  ( A. z ( z  <Q 
y  ->  z  e.  x )  /\  E. z  e.  x  y  <Q  z ) ) }
241, 23wceq 1369 1  wff  P.  =  { x  |  (
( (/)  C.  x  /\  x  C.  Q. )  /\  A. y  e.  x  ( A. z ( z 
<Q  y  ->  z  e.  x )  /\  E. z  e.  x  y  <Q  z ) ) }
Colors of variables: wff setvar class
This definition is referenced by:  npex  9155  elnp  9156  elnpi  9157
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