Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sltsgn2 Structured version   Unicode version

Theorem sltsgn2 27822
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 27816 . 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 5720 . . . 4  |-  ( A `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  e. 
_V
3 fvex 5720 . . . 4  |-  ( B `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  e. 
_V
42, 3brtp 27578 . . 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 1281 . . 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 964    = wceq 1369    e. wcel 1756    =/= wne 2620   {crab 2738   (/)c0 3656   {ctp 3900   <.cop 3902   |^|cint 4147   class class class wbr 4311   Oncon0 4738   ` cfv 5437   1oc1o 6932   2oc2o 6933   Nocsur 27800   <scslt 27801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-br 4312  df-opab 4370  df-tr 4405  df-eprel 4651  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-suc 4744  df-iota 5400  df-fv 5445  df-1o 6939  df-2o 6940  df-slt 27804
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator