MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfuni2 Unicode version

Theorem dfuni2 3845
Description: Alternate definition of class union. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
dfuni2  |-  U. A  =  { x  |  E. y  e.  A  x  e.  y }
Distinct variable group:    x, y, A

Proof of Theorem dfuni2
StepHypRef Expression
1 df-uni 3844 . 2  |-  U. A  =  { x  |  E. y ( x  e.  y  /\  y  e.  A ) }
2 exancom 1576 . . . 4  |-  ( E. y ( x  e.  y  /\  y  e.  A )  <->  E. y
( y  e.  A  /\  x  e.  y
) )
3 df-rex 2562 . . . 4  |-  ( E. y  e.  A  x  e.  y  <->  E. y
( y  e.  A  /\  x  e.  y
) )
42, 3bitr4i 243 . . 3  |-  ( E. y ( x  e.  y  /\  y  e.  A )  <->  E. y  e.  A  x  e.  y )
54abbii 2408 . 2  |-  { x  |  E. y ( x  e.  y  /\  y  e.  A ) }  =  { x  |  E. y  e.  A  x  e.  y }
61, 5eqtri 2316 1  |-  U. A  =  { x  |  E. y  e.  A  x  e.  y }
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   U.cuni 3843
This theorem is referenced by:  nfuni  3849  nfunid  3850  unieq  3852  uniiun  3971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-rex 2562  df-uni 3844
  Copyright terms: Public domain W3C validator