MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unrab Unicode version

Theorem unrab 3439
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 2552 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
2 df-rab 2552 . . 3  |-  { x  e.  A  |  ps }  =  { x  |  ( x  e.  A  /\  ps ) }
31, 2uneq12i 3327 . 2  |-  ( { x  e.  A  |  ph }  u.  { x  e.  A  |  ps } )  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  {
x  |  ( x  e.  A  /\  ps ) } )
4 df-rab 2552 . . 3  |-  { x  e.  A  |  ( ph  \/  ps ) }  =  { x  |  ( x  e.  A  /\  ( ph  \/  ps ) ) }
5 unab 3435 . . . 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 2395 . . . 4  |-  { x  |  ( x  e.  A  /\  ( ph  \/  ps ) ) }  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  A  /\  ps ) ) }
85, 7eqtr4i 2306 . . 3  |-  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  A  /\  ps ) } )  =  {
x  |  ( x  e.  A  /\  ( ph  \/  ps ) ) }
94, 8eqtr4i 2306 . 2  |-  { x  e.  A  |  ( ph  \/  ps ) }  =  ( { x  |  ( x  e.  A  /\  ph ) }  u.  { x  |  ( x  e.  A  /\  ps ) } )
103, 9eqtr4i 2306 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 1623    e. wcel 1684   {cab 2269   {crab 2547    u. cun 3150
This theorem is referenced by:  rabxm  3477  kmlem3  7778  hashbclem  11390  phiprmpw  12844  efgsfo  15048  mumul  20419  ppiub  20443  lgsquadlem2  20594  hasheuni  23453  measvuni  23542  subfacp1lem6  23716  lineunray  24770  orrabdioph  26861  dsmmacl  27207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-un 3157
  Copyright terms: Public domain W3C validator