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Theorem uniprg 3844
Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)
Assertion
Ref Expression
uniprg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U. { A ,  B }  =  ( A  u.  B )
)
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem uniprg
StepHypRef Expression
1 preq1 3708 . . . 4  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
21unieqd 3840 . . 3  |-  ( x  =  A  ->  U. {
x ,  y }  =  U. { A ,  y } )
3 uneq1 3324 . . 3  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
42, 3eqeq12d 2299 . 2  |-  ( x  =  A  ->  ( U. { x ,  y }  =  ( x  u.  y )  <->  U. { A ,  y }  =  ( A  u.  y
) ) )
5 preq2 3709 . . . 4  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
65unieqd 3840 . . 3  |-  ( y  =  B  ->  U. { A ,  y }  =  U. { A ,  B } )
7 uneq2 3325 . . 3  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
86, 7eqeq12d 2299 . 2  |-  ( y  =  B  ->  ( U. { A ,  y }  =  ( A  u.  y )  <->  U. { A ,  B }  =  ( A  u.  B ) ) )
9 vex 2793 . . 3  |-  x  e. 
_V
10 vex 2793 . . 3  |-  y  e. 
_V
119, 10unipr 3843 . 2  |-  U. {
x ,  y }  =  ( x  u.  y )
124, 8, 11vtocl2g 2849 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U. { A ,  B }  =  ( A  u.  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    u. cun 3152   {cpr 3643   U.cuni 3829
This theorem is referenced by:  wunun  8328  tskun  8404  gruun  8424  mrcun  13519  unopn  16644  indistopon  16733  uncon  17150  limcun  19240  sshjval3  21926  indispcon  23170  kelac2  26563
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-rex 2551  df-v 2792  df-un 3159  df-sn 3648  df-pr 3649  df-uni 3830
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