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Theorem uniprg 3751
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 3600 . . . 4  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
21unieqd 3747 . . 3  |-  ( x  =  A  ->  U. {
x ,  y }  =  U. { A ,  y } )
3 uneq1 3223 . . 3  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
42, 3eqeq12d 2154 . 2  |-  ( x  =  A  ->  ( U. { x ,  y }  =  ( x  u.  y )  <->  U. { A ,  y }  =  ( A  u.  y
) ) )
5 preq2 3601 . . . 4  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
65unieqd 3747 . . 3  |-  ( y  =  B  ->  U. { A ,  y }  =  U. { A ,  B } )
7 uneq2 3224 . . 3  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
86, 7eqeq12d 2154 . 2  |-  ( y  =  B  ->  ( U. { A ,  y }  =  ( A  u.  y )  <->  U. { A ,  B }  =  ( A  u.  B ) ) )
9 vex 2689 . . 3  |-  x  e. 
_V
10 vex 2689 . . 3  |-  y  e. 
_V
119, 10unipr 3750 . 2  |-  U. {
x ,  y }  =  ( x  u.  y )
124, 8, 11vtocl2g 2750 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U. { A ,  B }  =  ( A  u.  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480    u. cun 3069   {cpr 3528   U.cuni 3736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-uni 3737
This theorem is referenced by:  onun2  4406  unopn  12181
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