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Theorem unrab 3317
Description: Union of two restricted class abstractions. (Contributed by NM, 25-Mar-2004.)
Assertion
Ref Expression
unrab  |-  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  ps } )  =  {
x  e.  A  | 
( ph  \/  ps ) }

Proof of Theorem unrab
StepHypRef Expression
1 df-rab 2402 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-rab 2402 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
31, 2uneq12i 3198 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  ps } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  {
x  |  ( x  e.  A  /\  ps ) } )
4 df-rab 2402 . . 3  |-  { x  e.  A  |  ( ph  \/  ps ) }  =  { x  |  ( x  e.  A  /\  ( ph  \/  ps ) ) }
5 unab 3313 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps ) ) }
6 andi 792 . . . . 5  |-  ( ( x  e.  A  /\  ( ph  \/  ps )
)  <->  ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps )
) )
76abbii 2233 . . . 4  |-  { x  |  ( x  e.  A  /\  ( ph  \/  ps ) ) }  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps ) ) }
85, 7eqtr4i 2141 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( x  e.  A  /\  ( ph  \/  ps ) ) }
94, 8eqtr4i 2141 . 2  |-  { x  e.  A  |  ( ph  \/  ps ) }  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  A  /\  ps ) } )
103, 9eqtr4i 2141 1  |-  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  ps } )  =  {
x  e.  A  | 
( ph  \/  ps ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    \/ wo 682    = wceq 1316    e. wcel 1465   {cab 2103   {crab 2397    u. cun 3039
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-un 3045
This theorem is referenced by:  rabxmdc  3364  phiprmpw  11825  unennn  11837  znnen  11838
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