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Theorem sltsgn2 25617
Description: If  A < s B, then the sign of  B at the first place they differ is either undefined or  2o (Contributed by Scott Fenton, 4-Sep-2011.)
Assertion
Ref Expression
sltsgn2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  ( ( B `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  =  (/)  \/  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  2o ) ) )
Distinct variable groups:    A, k    B, k

Proof of Theorem sltsgn2
StepHypRef Expression
1 sltval2 25611 . 2  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B 
<->  ( A `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } ) { <. 1o ,  (/)
>. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } ) ) )
2 fvex 5742 . . . 4  |-  ( A `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  e. 
_V
3 fvex 5742 . . . 4  |-  ( B `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  e. 
_V
42, 3brtp 25372 . . 3  |-  ( ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  <->  ( ( ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  1o  /\  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/) )  \/  (
( A `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  1o  /\  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  2o )  \/  ( ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/)  /\  ( B `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  2o ) ) )
5 orc 375 . . . . 5  |-  ( ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/)  ->  ( ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/)  \/  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  2o ) )
65adantl 453 . . . 4  |-  ( ( ( A `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  1o  /\  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/) )  ->  (
( B `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  (/)  \/  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  2o ) )
7 olc 374 . . . . 5  |-  ( ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  2o  ->  (
( B `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  (/)  \/  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  2o ) )
87adantl 453 . . . 4  |-  ( ( ( A `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  1o  /\  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  2o )  -> 
( ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/)  \/  ( B `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  2o ) )
97adantl 453 . . . 4  |-  ( ( ( A `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  (/)  /\  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  2o )  -> 
( ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/)  \/  ( B `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  2o ) )
106, 8, 93jaoi 1247 . . 3  |-  ( ( ( ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  1o 
/\  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/) )  \/  ( ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  1o  /\  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  2o )  \/  ( ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/)  /\  ( B `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  2o ) )  ->  ( ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/)  \/  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  2o ) )
114, 10sylbi 188 . 2  |-  ( ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } ) { <. 1o ,  (/) >. ,  <. 1o ,  2o >. ,  <. (/) ,  2o >. }  ( B `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  ->  ( ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/)  \/  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  2o ) )
121, 11syl6bi 220 1  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  ( ( B `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  =  (/)  \/  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  2o ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725    =/= wne 2599   {crab 2709   (/)c0 3628   {ctp 3816   <.cop 3817   |^|cint 4050   class class class wbr 4212   Oncon0 4581   ` cfv 5454   1oc1o 6717   2oc2o 6718   Nocsur 25595   < scslt 25596
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587  df-iota 5418  df-fv 5462  df-1o 6724  df-2o 6725  df-slt 25599
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