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Definition df-ltxr 8868
Description: Define 'less than' on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. Note that in our postulates for complex numbers,  <RR is primitive and not necessarily a relation on  RR. (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
df-ltxr  |-  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { 
-oo } )  X.  {  +oo } )  u.  ( {  -oo }  X.  RR ) ) )
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-ltxr
StepHypRef Expression
1 clt 8863 . 2  class  <
2 vx . . . . . . 7  set  x
32cv 1623 . . . . . 6  class  x
4 cr 8732 . . . . . 6  class  RR
53, 4wcel 1685 . . . . 5  wff  x  e.  RR
6 vy . . . . . . 7  set  y
76cv 1623 . . . . . 6  class  y
87, 4wcel 1685 . . . . 5  wff  y  e.  RR
9 cltrr 8737 . . . . . 6  class  <RR
103, 7, 9wbr 4025 . . . . 5  wff  x  <RR  y
115, 8, 10w3a 936 . . . 4  wff  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )
1211, 2, 6copab 4078 . . 3  class  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }
13 cmnf 8861 . . . . . . 7  class  -oo
1413csn 3642 . . . . . 6  class  {  -oo }
154, 14cun 3152 . . . . 5  class  ( RR  u.  {  -oo }
)
16 cpnf 8860 . . . . . 6  class  +oo
1716csn 3642 . . . . 5  class  {  +oo }
1815, 17cxp 4687 . . . 4  class  ( ( RR  u.  {  -oo } )  X.  {  +oo } )
1914, 4cxp 4687 . . . 4  class  ( { 
-oo }  X.  RR )
2018, 19cun 3152 . . 3  class  ( ( ( RR  u.  {  -oo } )  X.  {  +oo } )  u.  ( {  -oo }  X.  RR ) )
2112, 20cun 3152 . 2  class  ( {
<. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { 
-oo } )  X.  {  +oo } )  u.  ( {  -oo }  X.  RR ) ) )
221, 21wceq 1624 1  wff  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { 
-oo } )  X.  {  +oo } )  u.  ( {  -oo }  X.  RR ) ) )
Colors of variables: wff set class
This definition is referenced by:  ltrelxr  8882  ltxrlt  8889  ltxr  10453
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