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Theorem sltval 24859
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 2418 . . . . 5  |-  ( f  =  A  ->  (
f  e.  No  <->  A  e.  No ) )
21anbi1d 685 . . . 4  |-  ( f  =  A  ->  (
( f  e.  No  /\  g  e.  No )  <-> 
( A  e.  No  /\  g  e.  No ) ) )
3 fveq1 5604 . . . . . . . 8  |-  ( f  =  A  ->  (
f `  y )  =  ( A `  y ) )
43eqeq1d 2366 . . . . . . 7  |-  ( f  =  A  ->  (
( f `  y
)  =  ( g `
 y )  <->  ( A `  y )  =  ( g `  y ) ) )
54ralbidv 2639 . . . . . 6  |-  ( f  =  A  ->  ( A. y  e.  x  ( f `  y
)  =  ( g `
 y )  <->  A. y  e.  x  ( A `  y )  =  ( g `  y ) ) )
6 fveq1 5604 . . . . . . 7  |-  ( f  =  A  ->  (
f `  x )  =  ( A `  x ) )
76breq1d 4112 . . . . . 6  |-  ( f  =  A  ->  (
( f `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x )  <->  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) ) )
85, 7anbi12d 691 . . . . 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 2640 . . . 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 691 . . 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 2418 . . . . 5  |-  ( g  =  B  ->  (
g  e.  No  <->  B  e.  No ) )
1211anbi2d 684 . . . 4  |-  ( g  =  B  ->  (
( A  e.  No  /\  g  e.  No )  <-> 
( A  e.  No  /\  B  e.  No ) ) )
13 fveq1 5604 . . . . . . . 8  |-  ( g  =  B  ->  (
g `  y )  =  ( B `  y ) )
1413eqeq2d 2369 . . . . . . 7  |-  ( g  =  B  ->  (
( A `  y
)  =  ( g `
 y )  <->  ( A `  y )  =  ( B `  y ) ) )
1514ralbidv 2639 . . . . . 6  |-  ( g  =  B  ->  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  <->  A. y  e.  x  ( A `  y )  =  ( B `  y ) ) )
16 fveq1 5604 . . . . . . 7  |-  ( g  =  B  ->  (
g `  x )  =  ( B `  x ) )
1716breq2d 4114 . . . . . 6  |-  ( g  =  B  ->  (
( A `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x )  <->  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) )
1815, 17anbi12d 691 . . . . 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 2640 . . . 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 691 . . 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 24856 . . 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 4363 . 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 850 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 176    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   E.wrex 2620   (/)c0 3531   {ctp 3718   <.cop 3719   class class class wbr 4102   Oncon0 4471   ` cfv 5334   1oc1o 6556   2oc2o 6557   Nocsur 24852   < scslt 24853
This theorem is referenced by:  sltval2  24868  sltres  24876  nodense  24901  nobndup  24912  nobnddown  24913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-iota 5298  df-fv 5342  df-slt 24856
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