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Theorem unrab 3452
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 2565 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-rab 2565 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
31, 2uneq12i 3340 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  ps } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  {
x  |  ( x  e.  A  /\  ps ) } )
4 df-rab 2565 . . 3  |-  { x  e.  A  |  ( ph  \/  ps ) }  =  { x  |  ( x  e.  A  /\  ( ph  \/  ps ) ) }
5 unab 3448 . . . 4  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps ) ) }
6 andi 837 . . . . 5  |-  ( ( x  e.  A  /\  ( ph  \/  ps )
)  <->  ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps )
) )
76abbii 2408 . . . 4  |-  { x  |  ( x  e.  A  /\  ( ph  \/  ps ) ) }  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps ) ) }
85, 7eqtr4i 2319 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( x  e.  A  /\  ( ph  \/  ps ) ) }
94, 8eqtr4i 2319 . 2  |-  { x  e.  A  |  ( ph  \/  ps ) }  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  A  /\  ps ) } )
103, 9eqtr4i 2319 1  |-  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  ps } )  =  {
x  e.  A  | 
( ph  \/  ps ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   {crab 2560    u. cun 3163
This theorem is referenced by:  rabxm  3490  kmlem3  7794  hashbclem  11406  phiprmpw  12860  efgsfo  15064  mumul  20435  ppiub  20459  lgsquadlem2  20610  hasheuni  23468  measvuni  23557  subfacp1lem6  23731  lineunray  24842  itg2addnclem2  25004  itgaddnclem2  25010  iblabsnclem  25014  orrabdioph  26964  dsmmacl  27310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-un 3170
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