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Theorem unipr 4225
Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)
Hypotheses
Ref Expression
unipr.1  |-  A  e. 
_V
unipr.2  |-  B  e. 
_V
Assertion
Ref Expression
unipr  |-  U. { A ,  B }  =  ( A  u.  B )

Proof of Theorem unipr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.43 1756 . . . 4  |-  ( E. y ( ( x  e.  y  /\  y  =  A )  \/  (
x  e.  y  /\  y  =  B )
)  <->  ( E. y
( x  e.  y  /\  y  =  A )  \/  E. y
( x  e.  y  /\  y  =  B ) ) )
2 vex 3060 . . . . . . . 8  |-  y  e. 
_V
32elpr 3998 . . . . . . 7  |-  ( y  e.  { A ,  B }  <->  ( y  =  A  \/  y  =  B ) )
43anbi2i 705 . . . . . 6  |-  ( ( x  e.  y  /\  y  e.  { A ,  B } )  <->  ( x  e.  y  /\  (
y  =  A  \/  y  =  B )
) )
5 andi 883 . . . . . 6  |-  ( ( x  e.  y  /\  ( y  =  A  \/  y  =  B ) )  <->  ( (
x  e.  y  /\  y  =  A )  \/  ( x  e.  y  /\  y  =  B ) ) )
64, 5bitri 257 . . . . 5  |-  ( ( x  e.  y  /\  y  e.  { A ,  B } )  <->  ( (
x  e.  y  /\  y  =  A )  \/  ( x  e.  y  /\  y  =  B ) ) )
76exbii 1729 . . . 4  |-  ( E. y ( x  e.  y  /\  y  e. 
{ A ,  B } )  <->  E. y
( ( x  e.  y  /\  y  =  A )  \/  (
x  e.  y  /\  y  =  B )
) )
8 unipr.1 . . . . . . 7  |-  A  e. 
_V
98clel3 3189 . . . . . 6  |-  ( x  e.  A  <->  E. y
( y  =  A  /\  x  e.  y ) )
10 exancom 1733 . . . . . 6  |-  ( E. y ( y  =  A  /\  x  e.  y )  <->  E. y
( x  e.  y  /\  y  =  A ) )
119, 10bitri 257 . . . . 5  |-  ( x  e.  A  <->  E. y
( x  e.  y  /\  y  =  A ) )
12 unipr.2 . . . . . . 7  |-  B  e. 
_V
1312clel3 3189 . . . . . 6  |-  ( x  e.  B  <->  E. y
( y  =  B  /\  x  e.  y ) )
14 exancom 1733 . . . . . 6  |-  ( E. y ( y  =  B  /\  x  e.  y )  <->  E. y
( x  e.  y  /\  y  =  B ) )
1513, 14bitri 257 . . . . 5  |-  ( x  e.  B  <->  E. y
( x  e.  y  /\  y  =  B ) )
1611, 15orbi12i 528 . . . 4  |-  ( ( x  e.  A  \/  x  e.  B )  <->  ( E. y ( x  e.  y  /\  y  =  A )  \/  E. y ( x  e.  y  /\  y  =  B ) ) )
171, 7, 163bitr4ri 286 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  <->  E. y ( x  e.  y  /\  y  e. 
{ A ,  B } ) )
1817abbii 2578 . 2  |-  { x  |  ( x  e.  A  \/  x  e.  B ) }  =  { x  |  E. y ( x  e.  y  /\  y  e. 
{ A ,  B } ) }
19 df-un 3421 . 2  |-  ( A  u.  B )  =  { x  |  ( x  e.  A  \/  x  e.  B ) }
20 df-uni 4213 . 2  |-  U. { A ,  B }  =  { x  |  E. y ( x  e.  y  /\  y  e. 
{ A ,  B } ) }
2118, 19, 203eqtr4ri 2495 1  |-  U. { A ,  B }  =  ( A  u.  B )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 374    /\ wa 375    = wceq 1455   E.wex 1674    e. wcel 1898   {cab 2448   _Vcvv 3057    u. cun 3414   {cpr 3982   U.cuni 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-un 3421  df-sn 3981  df-pr 3983  df-uni 4213
This theorem is referenced by:  uniprg  4226  unisn  4227  uniintsn  4286  uniop  4718  unex  6616  rankxplim  8376  mrcun  15577  indistps  20075  indistps2  20076  leordtval2  20277  ex-uni  25925  fouriersw  38133
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