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Theorem unipr 4235
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 1740 . . . 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 3090 . . . . . . . 8  |-  y  e. 
_V
32elpr 4020 . . . . . . 7  |-  ( y  e.  { A ,  B }  <->  ( y  =  A  \/  y  =  B ) )
43anbi2i 698 . . . . . 6  |-  ( ( x  e.  y  /\  y  e.  { A ,  B } )  <->  ( x  e.  y  /\  (
y  =  A  \/  y  =  B )
) )
5 andi 875 . . . . . 6  |-  ( ( x  e.  y  /\  ( y  =  A  \/  y  =  B ) )  <->  ( (
x  e.  y  /\  y  =  A )  \/  ( x  e.  y  /\  y  =  B ) ) )
64, 5bitri 252 . . . . 5  |-  ( ( x  e.  y  /\  y  e.  { A ,  B } )  <->  ( (
x  e.  y  /\  y  =  A )  \/  ( x  e.  y  /\  y  =  B ) ) )
76exbii 1714 . . . 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 3216 . . . . . 6  |-  ( x  e.  A  <->  E. y
( y  =  A  /\  x  e.  y ) )
10 exancom 1718 . . . . . 6  |-  ( E. y ( y  =  A  /\  x  e.  y )  <->  E. y
( x  e.  y  /\  y  =  A ) )
119, 10bitri 252 . . . . 5  |-  ( x  e.  A  <->  E. y
( x  e.  y  /\  y  =  A ) )
12 unipr.2 . . . . . . 7  |-  B  e. 
_V
1312clel3 3216 . . . . . 6  |-  ( x  e.  B  <->  E. y
( y  =  B  /\  x  e.  y ) )
14 exancom 1718 . . . . . 6  |-  ( E. y ( y  =  B  /\  x  e.  y )  <->  E. y
( x  e.  y  /\  y  =  B ) )
1513, 14bitri 252 . . . . 5  |-  ( x  e.  B  <->  E. y
( x  e.  y  /\  y  =  B ) )
1611, 15orbi12i 523 . . . 4  |-  ( ( x  e.  A  \/  x  e.  B )  <->  ( E. y ( x  e.  y  /\  y  =  A )  \/  E. y ( x  e.  y  /\  y  =  B ) ) )
171, 7, 163bitr4ri 281 . . 3  |-  ( ( x  e.  A  \/  x  e.  B )  <->  E. y ( x  e.  y  /\  y  e. 
{ A ,  B } ) )
1817abbii 2563 . 2  |-  { x  |  ( x  e.  A  \/  x  e.  B ) }  =  { x  |  E. y ( x  e.  y  /\  y  e. 
{ A ,  B } ) }
19 df-un 3447 . 2  |-  ( A  u.  B )  =  { x  |  ( x  e.  A  \/  x  e.  B ) }
20 df-uni 4223 . 2  |-  U. { A ,  B }  =  { x  |  E. y ( x  e.  y  /\  y  e. 
{ A ,  B } ) }
2118, 19, 203eqtr4ri 2469 1  |-  U. { A ,  B }  =  ( A  u.  B )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 369    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870   {cab 2414   _Vcvv 3087    u. cun 3440   {cpr 4004   U.cuni 4222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-un 3447  df-sn 4003  df-pr 4005  df-uni 4223
This theorem is referenced by:  uniprg  4236  unisn  4237  uniintsn  4296  uniop  4724  unex  6603  rankxplim  8349  mrcun  15479  indistps  19957  indistps2  19958  leordtval2  20159  ex-uni  25721  fouriersw  37662
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