Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  decidi Unicode version

Theorem decidi 11695
Description: Property of being decidable in another class. (Contributed by BJ, 19-Feb-2022.)
Assertion
Ref Expression
decidi  |-  ( A DECIDin  B  -> 
( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A
) ) )

Proof of Theorem decidi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-dcin 11694 . 2  |-  ( A DECIDin  B  <->  A. x  e.  B DECID  x  e.  A
)
2 df-dc 781 . . . 4  |-  (DECID  x  e.  A  <->  ( x  e.  A  \/  -.  x  e.  A ) )
32ralbii 2384 . . 3  |-  ( A. x  e.  B DECID  x  e.  A 
<-> 
A. x  e.  B  ( x  e.  A  \/  -.  x  e.  A
) )
4 eleq1 2150 . . . . 5  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
54notbid 627 . . . . 5  |-  ( x  =  X  ->  ( -.  x  e.  A  <->  -.  X  e.  A ) )
64, 5orbi12d 742 . . . 4  |-  ( x  =  X  ->  (
( x  e.  A  \/  -.  x  e.  A
)  <->  ( X  e.  A  \/  -.  X  e.  A ) ) )
76rspccv 2719 . . 3  |-  ( A. x  e.  B  (
x  e.  A  \/  -.  x  e.  A
)  ->  ( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A )
) )
83, 7sylbi 119 . 2  |-  ( A. x  e.  B DECID  x  e.  A  ->  ( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A )
) )
91, 8sylbi 119 1  |-  ( A DECIDin  B  -> 
( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 664  DECID wdc 780    = wceq 1289    e. wcel 1438   A.wral 2359   DECIDin wdcin 11693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-dcin 11694
This theorem is referenced by:  decidin  11697
  Copyright terms: Public domain W3C validator