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Definition df-reap 7639
Description: Define real apartness. Definition in Section 11.2.1 of [HoTT], p. (varies). Although # is an apartness relation on the reals (see df-ap 7646 for more discussion of apartness relations), for our purposes it is just a stepping stone to defining # which is an apartness relation on complex numbers. On the reals, # and # agree (apreap 7651). (Contributed by Jim Kingdon, 26-Jan-2020.)
Assertion
Ref Expression
df-reap  |- #  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  < 
y  \/  y  < 
x ) ) }
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-reap
StepHypRef Expression
1 creap 7638 . 2  class #
2 vx . . . . . . 7  setvar  x
32cv 1258 . . . . . 6  class  x
4 cr 6945 . . . . . 6  class  RR
53, 4wcel 1409 . . . . 5  wff  x  e.  RR
6 vy . . . . . . 7  setvar  y
76cv 1258 . . . . . 6  class  y
87, 4wcel 1409 . . . . 5  wff  y  e.  RR
95, 8wa 101 . . . 4  wff  ( x  e.  RR  /\  y  e.  RR )
10 clt 7118 . . . . . 6  class  <
113, 7, 10wbr 3791 . . . . 5  wff  x  < 
y
127, 3, 10wbr 3791 . . . . 5  wff  y  < 
x
1311, 12wo 639 . . . 4  wff  ( x  <  y  \/  y  <  x )
149, 13wa 101 . . 3  wff  ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  <  y  \/  y  <  x ) )
1514, 2, 6copab 3844 . 2  class  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  < 
y  \/  y  < 
x ) ) }
161, 15wceq 1259 1  wff #  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  < 
y  \/  y  < 
x ) ) }
Colors of variables: wff set class
This definition is referenced by:  reapval  7640
  Copyright terms: Public domain W3C validator