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Theorem sltval 29302
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 2539 . . . . 5  |-  ( f  =  A  ->  (
f  e.  No  <->  A  e.  No ) )
21anbi1d 704 . . . 4  |-  ( f  =  A  ->  (
( f  e.  No  /\  g  e.  No )  <-> 
( A  e.  No  /\  g  e.  No ) ) )
3 fveq1 5870 . . . . . . . 8  |-  ( f  =  A  ->  (
f `  y )  =  ( A `  y ) )
43eqeq1d 2469 . . . . . . 7  |-  ( f  =  A  ->  (
( f `  y
)  =  ( g `
 y )  <->  ( A `  y )  =  ( g `  y ) ) )
54ralbidv 2906 . . . . . 6  |-  ( f  =  A  ->  ( A. y  e.  x  ( f `  y
)  =  ( g `
 y )  <->  A. y  e.  x  ( A `  y )  =  ( g `  y ) ) )
6 fveq1 5870 . . . . . . 7  |-  ( f  =  A  ->  (
f `  x )  =  ( A `  x ) )
76breq1d 4462 . . . . . 6  |-  ( f  =  A  ->  (
( f `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x )  <->  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x ) ) )
85, 7anbi12d 710 . . . . 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 2978 . . . 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 710 . . 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 2539 . . . . 5  |-  ( g  =  B  ->  (
g  e.  No  <->  B  e.  No ) )
1211anbi2d 703 . . . 4  |-  ( g  =  B  ->  (
( A  e.  No  /\  g  e.  No )  <-> 
( A  e.  No  /\  B  e.  No ) ) )
13 fveq1 5870 . . . . . . . 8  |-  ( g  =  B  ->  (
g `  y )  =  ( B `  y ) )
1413eqeq2d 2481 . . . . . . 7  |-  ( g  =  B  ->  (
( A `  y
)  =  ( g `
 y )  <->  ( A `  y )  =  ( B `  y ) ) )
1514ralbidv 2906 . . . . . 6  |-  ( g  =  B  ->  ( A. y  e.  x  ( A `  y )  =  ( g `  y )  <->  A. y  e.  x  ( A `  y )  =  ( B `  y ) ) )
16 fveq1 5870 . . . . . . 7  |-  ( g  =  B  ->  (
g `  x )  =  ( B `  x ) )
1716breq2d 4464 . . . . . 6  |-  ( g  =  B  ->  (
( A `  x
) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( g `  x )  <->  ( A `  x ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  x ) ) )
1815, 17anbi12d 710 . . . . 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 2978 . . . 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 710 . . 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 29299 . . 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 4771 . 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 878 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   (/)c0 3790   {ctp 4036   <.cop 4038   class class class wbr 4452   Oncon0 4883   ` cfv 5593   1oc1o 7133   2oc2o 7134   Nocsur 29295   <scslt 29296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-iota 5556  df-fv 5601  df-slt 29299
This theorem is referenced by:  sltval2  29311  sltres  29319  nodense  29344  nobndup  29355  nobnddown  29356
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