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Definition df-slt 24369
Description: Next, we introduce surreal less-than, a comparison relationship over the surreals by lexicographically ordering them. (Contributed by Scott Fenton, 9-Jun-2011.)
Assertion
Ref Expression
df-slt  |-  < s  =  { <. f ,  g
>.  |  ( (
f  e.  No  /\  g  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( f `  y
)  =  ( g `
 y )  /\  ( f `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) ) ) }
Distinct variable group:    f, g, x, y

Detailed syntax breakdown of Definition df-slt
StepHypRef Expression
1 cslt 24366 . 2  class  < s
2 vf . . . . . . 7  set  f
32cv 1631 . . . . . 6  class  f
4 csur 24365 . . . . . 6  class  No
53, 4wcel 1696 . . . . 5  wff  f  e.  No
6 vg . . . . . . 7  set  g
76cv 1631 . . . . . 6  class  g
87, 4wcel 1696 . . . . 5  wff  g  e.  No
95, 8wa 358 . . . 4  wff  ( f  e.  No  /\  g  e.  No )
10 vy . . . . . . . . . 10  set  y
1110cv 1631 . . . . . . . . 9  class  y
1211, 3cfv 5271 . . . . . . . 8  class  ( f `
 y )
1311, 7cfv 5271 . . . . . . . 8  class  ( g `
 y )
1412, 13wceq 1632 . . . . . . 7  wff  ( f `
 y )  =  ( g `  y
)
15 vx . . . . . . . 8  set  x
1615cv 1631 . . . . . . 7  class  x
1714, 10, 16wral 2556 . . . . . 6  wff  A. y  e.  x  ( f `  y )  =  ( g `  y )
1816, 3cfv 5271 . . . . . . 7  class  ( f `
 x )
1916, 7cfv 5271 . . . . . . 7  class  ( g `
 x )
20 c1o 6488 . . . . . . . . 9  class  1o
21 c0 3468 . . . . . . . . 9  class  (/)
2220, 21cop 3656 . . . . . . . 8  class  <. 1o ,  (/)
>.
23 c2o 6489 . . . . . . . . 9  class  2o
2420, 23cop 3656 . . . . . . . 8  class  <. 1o ,  2o >.
2521, 23cop 3656 . . . . . . . 8  class  <. (/) ,  2o >.
2622, 24, 25ctp 3655 . . . . . . 7  class  { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }
2718, 19, 26wbr 4039 . . . . . 6  wff  ( f `
 x ) {
<. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/)
,  2o >. }  (
g `  x )
2817, 27wa 358 . . . . 5  wff  ( A. y  e.  x  (
f `  y )  =  ( g `  y )  /\  (
f `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x
) )
29 con0 4408 . . . . 5  class  On
3028, 15, 29wrex 2557 . . . 4  wff  E. x  e.  On  ( A. y  e.  x  ( f `  y )  =  ( g `  y )  /\  ( f `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) )
319, 30wa 358 . . 3  wff  ( ( f  e.  No  /\  g  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( f `  y
)  =  ( g `
 y )  /\  ( f `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) ) )
3231, 2, 6copab 4092 . 2  class  { <. f ,  g >.  |  ( ( f  e.  No  /\  g  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( f `  y )  =  ( g `  y )  /\  ( f `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) ) ) }
331, 32wceq 1632 1  wff  < s  =  { <. f ,  g
>.  |  ( (
f  e.  No  /\  g  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( f `  y
)  =  ( g `
 y )  /\  ( f `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) ) ) }
Colors of variables: wff set class
This definition is referenced by:  sltval  24372  sltso  24394
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