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Theorem sltsgn2 29349
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 29343 . 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 5882 . . . 4  |-  ( A `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  e. 
_V
3 fvex 5882 . . . 4  |-  ( B `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  e. 
_V
42, 3brtp 29105 . . 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 385 . . . . 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 466 . . . 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 384 . . . . 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 466 . . . 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 466 . . . 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 1291 . . 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 195 . 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 228 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 setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2821   (/)c0 3790   {ctp 4037   <.cop 4039   |^|cint 4288   class class class wbr 4453   Oncon0 4884   ` cfv 5594   1oc1o 7135   2oc2o 7136   Nocsur 29327   <scslt 29328
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-suc 4890  df-iota 5557  df-fv 5602  df-1o 7142  df-2o 7143  df-slt 29331
This theorem is referenced by: (None)
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