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Theorem unopab 4111
Description: Union of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
unopab  |-  ( {
<. x ,  y >.  |  ph }  u.  { <. x ,  y >.  |  ps } )  =  { <. x ,  y
>.  |  ( ph  \/  ps ) }

Proof of Theorem unopab
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 unab 3448 . . 3  |-  ( { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) }  u.  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ps ) } )  =  { z  |  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  \/  E. x E. y ( z  =  <. x ,  y
>.  /\  ps ) ) }
2 19.43 1595 . . . . 5  |-  ( E. x ( E. y
( z  =  <. x ,  y >.  /\  ph )  \/  E. y
( z  =  <. x ,  y >.  /\  ps ) )  <->  ( E. x E. y ( z  =  <. x ,  y
>.  /\  ph )  \/ 
E. x E. y
( z  =  <. x ,  y >.  /\  ps ) ) )
3 andi 837 . . . . . . . 8  |-  ( ( z  =  <. x ,  y >.  /\  ( ph  \/  ps ) )  <-> 
( ( z  = 
<. x ,  y >.  /\  ph )  \/  (
z  =  <. x ,  y >.  /\  ps ) ) )
43exbii 1572 . . . . . . 7  |-  ( E. y ( z  = 
<. x ,  y >.  /\  ( ph  \/  ps ) )  <->  E. y
( ( z  = 
<. x ,  y >.  /\  ph )  \/  (
z  =  <. x ,  y >.  /\  ps ) ) )
5 19.43 1595 . . . . . . 7  |-  ( E. y ( ( z  =  <. x ,  y
>.  /\  ph )  \/  ( z  =  <. x ,  y >.  /\  ps ) )  <->  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  \/  E. y ( z  = 
<. x ,  y >.  /\  ps ) ) )
64, 5bitr2i 241 . . . . . 6  |-  ( ( E. y ( z  =  <. x ,  y
>.  /\  ph )  \/ 
E. y ( z  =  <. x ,  y
>.  /\  ps ) )  <->  E. y ( z  = 
<. x ,  y >.  /\  ( ph  \/  ps ) ) )
76exbii 1572 . . . . 5  |-  ( E. x ( E. y
( z  =  <. x ,  y >.  /\  ph )  \/  E. y
( z  =  <. x ,  y >.  /\  ps ) )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ( ph  \/  ps ) ) )
82, 7bitr3i 242 . . . 4  |-  ( ( E. x E. y
( z  =  <. x ,  y >.  /\  ph )  \/  E. x E. y ( z  = 
<. x ,  y >.  /\  ps ) )  <->  E. x E. y ( z  = 
<. x ,  y >.  /\  ( ph  \/  ps ) ) )
98abbii 2408 . . 3  |-  { z  |  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ph )  \/  E. x E. y ( z  =  <. x ,  y
>.  /\  ps ) ) }  =  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ( ph  \/  ps ) ) }
101, 9eqtri 2316 . 2  |-  ( { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) }  u.  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ps ) } )  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ( ph  \/  ps ) ) }
11 df-opab 4094 . . 3  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
12 df-opab 4094 . . 3  |-  { <. x ,  y >.  |  ps }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ps ) }
1311, 12uneq12i 3340 . 2  |-  ( {
<. x ,  y >.  |  ph }  u.  { <. x ,  y >.  |  ps } )  =  ( { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }  u.  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ps ) } )
14 df-opab 4094 . 2  |-  { <. x ,  y >.  |  (
ph  \/  ps ) }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ( ph  \/  ps ) ) }
1510, 13, 143eqtr4i 2326 1  |-  ( {
<. x ,  y >.  |  ph }  u.  { <. x ,  y >.  |  ps } )  =  { <. x ,  y
>.  |  ( ph  \/  ps ) }
Colors of variables: wff set class
Syntax hints:    \/ wo 357    /\ wa 358   E.wex 1531    = wceq 1632   {cab 2282    u. cun 3163   <.cop 3656   {copab 4092
This theorem is referenced by:  xpundi  4757  xpundir  4758  cnvun  5102  coundi  5190  coundir  5191  mptun  5390  opsrtoslem1  16241  lgsquadlem3  20611
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-v 2803  df-un 3170  df-opab 4094
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