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Theorem unipr 3989
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 1612 . . . 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 2919 . . . . . . . 8  |-  y  e. 
_V
32elpr 3792 . . . . . . 7  |-  ( y  e.  { A ,  B }  <->  ( y  =  A  \/  y  =  B ) )
43anbi2i 676 . . . . . 6  |-  ( ( x  e.  y  /\  y  e.  { A ,  B } )  <->  ( x  e.  y  /\  (
y  =  A  \/  y  =  B )
) )
5 andi 838 . . . . . 6  |-  ( ( x  e.  y  /\  ( y  =  A  \/  y  =  B ) )  <->  ( (
x  e.  y  /\  y  =  A )  \/  ( x  e.  y  /\  y  =  B ) ) )
64, 5bitri 241 . . . . 5  |-  ( ( x  e.  y  /\  y  e.  { A ,  B } )  <->  ( (
x  e.  y  /\  y  =  A )  \/  ( x  e.  y  /\  y  =  B ) ) )
76exbii 1589 . . . 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 3034 . . . . . 6  |-  ( x  e.  A  <->  E. y
( y  =  A  /\  x  e.  y ) )
10 exancom 1593 . . . . . 6  |-  ( E. y ( y  =  A  /\  x  e.  y )  <->  E. y
( x  e.  y  /\  y  =  A ) )
119, 10bitri 241 . . . . 5  |-  ( x  e.  A  <->  E. y
( x  e.  y  /\  y  =  A ) )
12 unipr.2 . . . . . . 7  |-  B  e. 
_V
1312clel3 3034 . . . . . 6  |-  ( x  e.  B  <->  E. y
( y  =  B  /\  x  e.  y ) )
14 exancom 1593 . . . . . 6  |-  ( E. y ( y  =  B  /\  x  e.  y )  <->  E. y
( x  e.  y  /\  y  =  B ) )
1513, 14bitri 241 . . . . 5  |-  ( x  e.  B  <->  E. y
( x  e.  y  /\  y  =  B ) )
1611, 15orbi12i 508 . . . 4  |-  ( ( x  e.  A  \/  x  e.  B )  <->  ( E. y ( x  e.  y  /\  y  =  A )  \/  E. y ( x  e.  y  /\  y  =  B ) ) )
171, 7, 163bitr4ri 270 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  <->  E. y ( x  e.  y  /\  y  e. 
{ A ,  B } ) )
1817abbii 2516 . 2  |-  { x  |  ( x  e.  A  \/  x  e.  B ) }  =  { x  |  E. y ( x  e.  y  /\  y  e. 
{ A ,  B } ) }
19 df-un 3285 . 2  |-  ( A  u.  B )  =  { x  |  ( x  e.  A  \/  x  e.  B ) }
20 df-uni 3976 . 2  |-  U. { A ,  B }  =  { x  |  E. y ( x  e.  y  /\  y  e. 
{ A ,  B } ) }
2118, 19, 203eqtr4ri 2435 1  |-  U. { A ,  B }  =  ( A  u.  B )
Colors of variables: wff set class
Syntax hints:    \/ wo 358    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390   _Vcvv 2916    u. cun 3278   {cpr 3775   U.cuni 3975
This theorem is referenced by:  uniprg  3990  unisn  3991  uniintsn  4047  uniop  4419  unex  4666  rankxplim  7759  mrcun  13802  indistps  17030  indistps2  17031  leordtval2  17230  ex-uni  21687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285  df-sn 3780  df-pr 3781  df-uni 3976
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