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Theorem sltval 25607
Description: The value of the surreal less than relationship. (Contributed by Scott Fenton, 14-Jun-2011.)
Assertion
Ref Expression
sltval  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B 
<->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) ) )
Distinct variable groups:    x, A, y    x, B, y

Proof of Theorem sltval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2498 . . . . 5  |-  ( f  =  A  ->  (
f  e.  No  <->  A  e.  No ) )
21anbi1d 687 . . . 4  |-  ( f  =  A  ->  (
( f  e.  No  /\  g  e.  No )  <-> 
( A  e.  No  /\  g  e.  No ) ) )
3 fveq1 5730 . . . . . . . 8  |-  ( f  =  A  ->  (
f `  y )  =  ( A `  y ) )
43eqeq1d 2446 . . . . . . 7  |-  ( f  =  A  ->  (
( f `  y
)  =  ( g `
 y )  <->  ( A `  y )  =  ( g `  y ) ) )
54ralbidv 2727 . . . . . 6  |-  ( f  =  A  ->  ( A. y  e.  x  ( f `  y
)  =  ( g `
 y )  <->  A. y  e.  x  ( A `  y )  =  ( g `  y ) ) )
6 fveq1 5730 . . . . . . 7  |-  ( f  =  A  ->  (
f `  x )  =  ( A `  x ) )
76breq1d 4225 . . . . . 6  |-  ( f  =  A  ->  (
( f `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x )  <->  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) ) )
85, 7anbi12d 693 . . . . 5  |-  ( f  =  A  ->  (
( A. y  e.  x  ( f `  y )  =  ( g `  y )  /\  ( f `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) )  <->  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x
) ) ) )
98rexbidv 2728 . . . 4  |-  ( f  =  A  ->  ( E. x  e.  On  ( A. y  e.  x  ( f `  y
)  =  ( g `
 y )  /\  ( f `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) )  <->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) ) ) )
102, 9anbi12d 693 . . 3  |-  ( f  =  A  ->  (
( ( 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 ) ) )  <-> 
( ( A  e.  No  /\  g  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x
) ) ) ) )
11 eleq1 2498 . . . . 5  |-  ( g  =  B  ->  (
g  e.  No  <->  B  e.  No ) )
1211anbi2d 686 . . . 4  |-  ( g  =  B  ->  (
( A  e.  No  /\  g  e.  No )  <-> 
( A  e.  No  /\  B  e.  No ) ) )
13 fveq1 5730 . . . . . . . 8  |-  ( g  =  B  ->  (
g `  y )  =  ( B `  y ) )
1413eqeq2d 2449 . . . . . . 7  |-  ( g  =  B  ->  (
( A `  y
)  =  ( g `
 y )  <->  ( A `  y )  =  ( B `  y ) ) )
1514ralbidv 2727 . . . . . 6  |-  ( g  =  B  ->  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  <->  A. y  e.  x  ( A `  y )  =  ( B `  y ) ) )
16 fveq1 5730 . . . . . . 7  |-  ( g  =  B  ->  (
g `  x )  =  ( B `  x ) )
1716breq2d 4227 . . . . . 6  |-  ( g  =  B  ->  (
( A `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x )  <->  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) )
1815, 17anbi12d 693 . . . . 5  |-  ( g  =  B  ->  (
( A. y  e.  x  ( A `  y )  =  ( g `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) )  <->  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) ) )
1918rexbidv 2728 . . . 4  |-  ( g  =  B  ->  ( E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x
) )  <->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) ) )
2012, 19anbi12d 693 . . 3  |-  ( g  =  B  ->  (
( ( A  e.  No  /\  g  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x
) ) )  <->  ( ( A  e.  No  /\  B  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) ) ) )
21 df-slt 25604 . . 3  |-  < 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 ) ) ) }
2210, 20, 21brabg 4477 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B 
<->  ( ( A  e.  No  /\  B  e.  No )  /\  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) ) ) )
2322bianabs 852 1  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B 
<->  E. x  e.  On  ( A. y  e.  x  ( A `  y )  =  ( B `  y )  /\  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   (/)c0 3630   {ctp 3818   <.cop 3819   class class class wbr 4215   Oncon0 4584   ` cfv 5457   1oc1o 6720   2oc2o 6721   Nocsur 25600   < scslt 25601
This theorem is referenced by:  sltval2  25616  sltres  25624  nodense  25649  nobndup  25660  nobnddown  25661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-iota 5421  df-fv 5465  df-slt 25604
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