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Theorem unipr 4253
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 1665 . . . 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 3111 . . . . . . . 8  |-  y  e. 
_V
32elpr 4040 . . . . . . 7  |-  ( y  e.  { A ,  B }  <->  ( y  =  A  \/  y  =  B ) )
43anbi2i 694 . . . . . 6  |-  ( ( x  e.  y  /\  y  e.  { A ,  B } )  <->  ( x  e.  y  /\  (
y  =  A  \/  y  =  B )
) )
5 andi 863 . . . . . 6  |-  ( ( x  e.  y  /\  ( y  =  A  \/  y  =  B ) )  <->  ( (
x  e.  y  /\  y  =  A )  \/  ( x  e.  y  /\  y  =  B ) ) )
64, 5bitri 249 . . . . 5  |-  ( ( x  e.  y  /\  y  e.  { A ,  B } )  <->  ( (
x  e.  y  /\  y  =  A )  \/  ( x  e.  y  /\  y  =  B ) ) )
76exbii 1639 . . . 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 3237 . . . . . 6  |-  ( x  e.  A  <->  E. y
( y  =  A  /\  x  e.  y ) )
10 exancom 1643 . . . . . 6  |-  ( E. y ( y  =  A  /\  x  e.  y )  <->  E. y
( x  e.  y  /\  y  =  A ) )
119, 10bitri 249 . . . . 5  |-  ( x  e.  A  <->  E. y
( x  e.  y  /\  y  =  A ) )
12 unipr.2 . . . . . . 7  |-  B  e. 
_V
1312clel3 3237 . . . . . 6  |-  ( x  e.  B  <->  E. y
( y  =  B  /\  x  e.  y ) )
14 exancom 1643 . . . . . 6  |-  ( E. y ( y  =  B  /\  x  e.  y )  <->  E. y
( x  e.  y  /\  y  =  B ) )
1513, 14bitri 249 . . . . 5  |-  ( x  e.  B  <->  E. y
( x  e.  y  /\  y  =  B ) )
1611, 15orbi12i 521 . . . 4  |-  ( ( x  e.  A  \/  x  e.  B )  <->  ( E. y ( x  e.  y  /\  y  =  A )  \/  E. y ( x  e.  y  /\  y  =  B ) ) )
171, 7, 163bitr4ri 278 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  <->  E. y ( x  e.  y  /\  y  e. 
{ A ,  B } ) )
1817abbii 2596 . 2  |-  { x  |  ( x  e.  A  \/  x  e.  B ) }  =  { x  |  E. y ( x  e.  y  /\  y  e. 
{ A ,  B } ) }
19 df-un 3476 . 2  |-  ( A  u.  B )  =  { x  |  ( x  e.  A  \/  x  e.  B ) }
20 df-uni 4241 . 2  |-  U. { A ,  B }  =  { x  |  E. y ( x  e.  y  /\  y  e. 
{ A ,  B } ) }
2118, 19, 203eqtr4ri 2502 1  |-  U. { A ,  B }  =  ( A  u.  B )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    = wceq 1374   E.wex 1591    e. wcel 1762   {cab 2447   _Vcvv 3108    u. cun 3469   {cpr 4024   U.cuni 4240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-v 3110  df-un 3476  df-sn 4023  df-pr 4025  df-uni 4241
This theorem is referenced by:  uniprg  4254  unisn  4255  uniintsn  4314  uniop  4745  unex  6575  rankxplim  8288  mrcun  14868  indistps  19273  indistps2  19274  leordtval2  19474  ex-uni  24812  fouriersw  31489
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