ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rabun2 Unicode version

Theorem rabun2 3259
Description: Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)
Assertion
Ref Expression
rabun2  |-  { x  e.  ( A  u.  B
)  |  ph }  =  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph } )

Proof of Theorem rabun2
StepHypRef Expression
1 df-rab 2362 . 2  |-  { x  e.  ( A  u.  B
)  |  ph }  =  { x  |  ( x  e.  ( A  u.  B )  /\  ph ) }
2 df-rab 2362 . . . 4  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
3 df-rab 2362 . . . 4  |-  { x  e.  B  |  ph }  =  { x  |  ( x  e.  B  /\  ph ) }
42, 3uneq12i 3134 . . 3  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph }
)  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )
5 elun 3123 . . . . . . 7  |-  ( x  e.  ( A  u.  B )  <->  ( x  e.  A  \/  x  e.  B ) )
65anbi1i 446 . . . . . 6  |-  ( ( x  e.  ( A  u.  B )  /\  ph )  <->  ( ( x  e.  A  \/  x  e.  B )  /\  ph ) )
7 andir 766 . . . . . 6  |-  ( ( ( x  e.  A  \/  x  e.  B
)  /\  ph )  <->  ( (
x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) )
86, 7bitri 182 . . . . 5  |-  ( ( x  e.  ( A  u.  B )  /\  ph )  <->  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph )
) )
98abbii 2198 . . . 4  |-  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) }
10 unab 3247 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  ph ) ) }
119, 10eqtr4i 2106 . . 3  |-  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  B  /\  ph ) } )
124, 11eqtr4i 2106 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph }
)  =  { x  |  ( x  e.  ( A  u.  B
)  /\  ph ) }
131, 12eqtr4i 2106 1  |-  { x  e.  ( A  u.  B
)  |  ph }  =  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  ph } )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    \/ wo 662    = wceq 1285    e. wcel 1434   {cab 2069   {crab 2357    u. cun 2980
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-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-v 2612  df-un 2986
This theorem is referenced by:  ssfirab  6476
  Copyright terms: Public domain W3C validator