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Theorem sltsgn1 25608
Description: If  A < s B, then the sign of  A at the first place they differ is either undefined or  1o (Contributed by Scott Fenton, 4-Sep-2011.)
Assertion
Ref Expression
sltsgn1  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  ( ( A `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  =  (/)  \/  ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  1o ) ) )
Distinct variable groups:    A, k    B, k

Proof of Theorem sltsgn1
StepHypRef Expression
1 sltval2 25603 . 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 5734 . . . 4  |-  ( A `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  e. 
_V
3 fvex 5734 . . . 4  |-  ( B `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  e. 
_V
42, 3brtp 25364 . . 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 olc 374 . . . . 5  |-  ( ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  1o  ->  (
( A `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  (/)  \/  ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  1o ) )
65adantr 452 . . . 4  |-  ( ( ( 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 ) } )  =  (/)  \/  ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  1o ) )
75adantr 452 . . . 4  |-  ( ( ( 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 ) } )  =  (/)  \/  ( A `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  1o ) )
8 orc 375 . . . . 5  |-  ( ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/)  ->  ( ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/)  \/  ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  1o ) )
98adantr 452 . . . 4  |-  ( ( ( A `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  (/)  /\  ( B `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  2o )  -> 
( ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/)  \/  ( A `  |^| { k  e.  On  | 
( A `  k
)  =/=  ( B `
 k ) } )  =  1o ) )
106, 7, 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 ) )  ->  ( ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  (/)  \/  ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  1o ) )
114, 10sylbi 188 . 2  |-  ( ( 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 ) } )  =  (/)  \/  ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  1o ) )
121, 11syl6bi 220 1  |-  ( ( A  e.  No  /\  B  e.  No )  ->  ( A < s B  ->  ( ( A `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  =  (/)  \/  ( A `  |^| { k  e.  On  |  ( A `  k )  =/=  ( B `  k ) } )  =  1o ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725    =/= wne 2598   {crab 2701   (/)c0 3620   {ctp 3808   <.cop 3809   |^|cint 4042   class class class wbr 4204   Oncon0 4573   ` cfv 5446   1oc1o 6709   2oc2o 6710   Nocsur 25587   < scslt 25588
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579  df-iota 5410  df-fv 5454  df-1o 6716  df-2o 6717  df-slt 25591
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