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Theorem uniprg 3663
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 )
)

Proof of Theorem uniprg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 3514 . . . 4  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
21unieqd 3659 . . 3  |-  ( x  =  A  ->  U. {
x ,  y }  =  U. { A ,  y } )
3 uneq1 3145 . . 3  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
42, 3eqeq12d 2102 . 2  |-  ( x  =  A  ->  ( U. { x ,  y }  =  ( x  u.  y )  <->  U. { A ,  y }  =  ( A  u.  y
) ) )
5 preq2 3515 . . . 4  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
65unieqd 3659 . . 3  |-  ( y  =  B  ->  U. { A ,  y }  =  U. { A ,  B } )
7 uneq2 3146 . . 3  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
86, 7eqeq12d 2102 . 2  |-  ( y  =  B  ->  ( U. { A ,  y }  =  ( A  u.  y )  <->  U. { A ,  B }  =  ( A  u.  B ) ) )
9 vex 2622 . . 3  |-  x  e. 
_V
10 vex 2622 . . 3  |-  y  e. 
_V
119, 10unipr 3662 . 2  |-  U. {
x ,  y }  =  ( x  u.  y )
124, 8, 11vtocl2g 2683 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U. { A ,  B }  =  ( A  u.  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438    u. cun 2995   {cpr 3442   U.cuni 3648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-uni 3649
This theorem is referenced by:  onun2  4297
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