MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniprg Unicode version

Theorem uniprg 3742
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
StepHypRef Expression
1 preq1 3610 . . . 4  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
21unieqd 3738 . . 3  |-  ( x  =  A  ->  U. {
x ,  y }  =  U. { A ,  y } )
3 uneq1 3232 . . 3  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
42, 3eqeq12d 2267 . 2  |-  ( x  =  A  ->  ( U. { x ,  y }  =  ( x  u.  y )  <->  U. { A ,  y }  =  ( A  u.  y
) ) )
5 preq2 3611 . . . 4  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
65unieqd 3738 . . 3  |-  ( y  =  B  ->  U. { A ,  y }  =  U. { A ,  B } )
7 uneq2 3233 . . 3  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
86, 7eqeq12d 2267 . 2  |-  ( y  =  B  ->  ( U. { A ,  y }  =  ( A  u.  y )  <->  U. { A ,  B }  =  ( A  u.  B ) ) )
9 vex 2730 . . 3  |-  x  e. 
_V
10 vex 2730 . . 3  |-  y  e. 
_V
119, 10unipr 3741 . 2  |-  U. {
x ,  y }  =  ( x  u.  y )
124, 8, 11vtocl2g 2785 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 1619    e. wcel 1621    u. cun 3076   {cpr 3545   U.cuni 3727
This theorem is referenced by:  wunun  8212  tskun  8288  gruun  8308  mrcun  13396  unopn  16481  indistopon  16570  uncon  16987  limcun  19077  sshjval3  21763  indispcon  22936  kelac2  26329
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-rex 2514  df-v 2729  df-un 3083  df-sn 3550  df-pr 3551  df-uni 3728
  Copyright terms: Public domain W3C validator