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

Theorem uniprg 4017
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 3870 . . . 4  |-  ( x  =  A  ->  { x ,  y }  =  { A ,  y } )
21unieqd 4013 . . 3  |-  ( x  =  A  ->  U. {
x ,  y }  =  U. { A ,  y } )
3 uneq1 3481 . . 3  |-  ( x  =  A  ->  (
x  u.  y )  =  ( A  u.  y ) )
42, 3eqeq12d 2444 . 2  |-  ( x  =  A  ->  ( U. { x ,  y }  =  ( x  u.  y )  <->  U. { A ,  y }  =  ( A  u.  y
) ) )
5 preq2 3871 . . . 4  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
65unieqd 4013 . . 3  |-  ( y  =  B  ->  U. { A ,  y }  =  U. { A ,  B } )
7 uneq2 3482 . . 3  |-  ( y  =  B  ->  ( A  u.  y )  =  ( A  u.  B ) )
86, 7eqeq12d 2444 . 2  |-  ( y  =  B  ->  ( U. { A ,  y }  =  ( A  u.  y )  <->  U. { A ,  B }  =  ( A  u.  B ) ) )
9 vex 2946 . . 3  |-  x  e. 
_V
10 vex 2946 . . 3  |-  y  e. 
_V
119, 10unipr 4016 . 2  |-  U. {
x ,  y }  =  ( x  u.  y )
124, 8, 11vtocl2g 3002 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  U. { A ,  B }  =  ( A  u.  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    u. cun 3305   {cpr 3802   U.cuni 4002
This theorem is referenced by:  wunun  8569  tskun  8645  gruun  8665  mrcun  13830  unopn  16959  indistopon  17048  uncon  17475  limcun  19765  sshjval3  22839  prsiga  24497  unelsiga  24500  measxun2  24547  measssd  24552  probun  24660  indispcon  24904  kelac2  27073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2411
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-rex 2698  df-v 2945  df-un 3312  df-sn 3807  df-pr 3808  df-uni 4003
  Copyright terms: Public domain W3C validator