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Theorem unrab 3268
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 2368 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-rab 2368 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
31, 2uneq12i 3150 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  ps } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  {
x  |  ( x  e.  A  /\  ps ) } )
4 df-rab 2368 . . 3  |-  { x  e.  A  |  ( ph  \/  ps ) }  =  { x  |  ( x  e.  A  /\  ( ph  \/  ps ) ) }
5 unab 3264 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps ) ) }
6 andi 767 . . . . 5  |-  ( ( x  e.  A  /\  ( ph  \/  ps )
)  <->  ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps )
) )
76abbii 2203 . . . 4  |-  { x  |  ( x  e.  A  /\  ( ph  \/  ps ) ) }  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps ) ) }
85, 7eqtr4i 2111 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( x  e.  A  /\  ( ph  \/  ps ) ) }
94, 8eqtr4i 2111 . 2  |-  { x  e.  A  |  ( ph  \/  ps ) }  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  A  /\  ps ) } )
103, 9eqtr4i 2111 1  |-  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  ps } )  =  {
x  e.  A  | 
( ph  \/  ps ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    \/ wo 664    = wceq 1289    e. wcel 1438   {cab 2074   {crab 2363    u. cun 2995
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 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-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-un 3001
This theorem is referenced by:  rabxmdc  3312  phiprmpw  11280  unennn  11292  znnen  11293
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