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

Theorem decidi 13016
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 13015 . 2  |-  ( A DECIDin  B  <->  A. x  e.  B DECID  x  e.  A
)
2 df-dc 820 . . . 4  |-  (DECID  x  e.  A  <->  ( x  e.  A  \/  -.  x  e.  A ) )
32ralbii 2441 . . 3  |-  ( A. x  e.  B DECID  x  e.  A 
<-> 
A. x  e.  B  ( x  e.  A  \/  -.  x  e.  A
) )
4 eleq1 2202 . . . . 5  |-  ( x  =  X  ->  (
x  e.  A  <->  X  e.  A ) )
54notbid 656 . . . . 5  |-  ( x  =  X  ->  ( -.  x  e.  A  <->  -.  X  e.  A ) )
64, 5orbi12d 782 . . . 4  |-  ( x  =  X  ->  (
( x  e.  A  \/  -.  x  e.  A
)  <->  ( X  e.  A  \/  -.  X  e.  A ) ) )
76rspccv 2786 . . 3  |-  ( A. x  e.  B  (
x  e.  A  \/  -.  x  e.  A
)  ->  ( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A )
) )
83, 7sylbi 120 . 2  |-  ( A. x  e.  B DECID  x  e.  A  ->  ( X  e.  B  ->  ( X  e.  A  \/  -.  X  e.  A )
) )
91, 8sylbi 120 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 697  DECID wdc 819    = wceq 1331    e. wcel 1480   A.wral 2416   DECIDin wdcin 13014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-dcin 13015
This theorem is referenced by:  decidin  13018
  Copyright terms: Public domain W3C validator