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Theorem sltsgn1 24386
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 24381 . 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 5555 . . . 4  |-  ( A `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  e. 
_V
3 fvex 5555 . . . 4  |-  ( B `
 |^| { k  e.  On  |  ( A `
 k )  =/=  ( B `  k
) } )  e. 
_V
42, 3brtp 24177 . . 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 373 . . . . 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 451 . . . 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 451 . . . 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 374 . . . . 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 451 . . . 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 1245 . . 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 187 . 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 219 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 357    /\ wa 358    \/ w3o 933    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   (/)c0 3468   {ctp 3655   <.cop 3656   |^|cint 3878   class class class wbr 4039   Oncon0 4408   ` cfv 5271   1oc1o 6488   2oc2o 6489   Nocsur 24365   < scslt 24366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-iota 5235  df-fv 5279  df-1o 6495  df-2o 6496  df-slt 24369
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