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

Theorem zfun 6492
Description: Axiom of Union expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
Assertion
Ref Expression
zfun  |-  E. x A. y ( E. x
( y  e.  x  /\  x  e.  z
)  ->  y  e.  x )
Distinct variable group:    x, y, z

Proof of Theorem zfun
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax-un 6491 . 2  |-  E. x A. y ( E. w
( y  e.  w  /\  w  e.  z
)  ->  y  e.  x )
2 elequ2 1831 . . . . . . 7  |-  ( w  =  x  ->  (
y  e.  w  <->  y  e.  x ) )
3 elequ1 1829 . . . . . . 7  |-  ( w  =  x  ->  (
w  e.  z  <->  x  e.  z ) )
42, 3anbi12d 708 . . . . . 6  |-  ( w  =  x  ->  (
( y  e.  w  /\  w  e.  z
)  <->  ( y  e.  x  /\  x  e.  z ) ) )
54cbvexv 2031 . . . . 5  |-  ( E. w ( y  e.  w  /\  w  e.  z )  <->  E. x
( y  e.  x  /\  x  e.  z
) )
65imbi1i 323 . . . 4  |-  ( ( E. w ( y  e.  w  /\  w  e.  z )  ->  y  e.  x )  <->  ( E. x ( y  e.  x  /\  x  e.  z )  ->  y  e.  x ) )
76albii 1648 . . 3  |-  ( A. y ( E. w
( y  e.  w  /\  w  e.  z
)  ->  y  e.  x )  <->  A. y
( E. x ( y  e.  x  /\  x  e.  z )  ->  y  e.  x ) )
87exbii 1675 . 2  |-  ( E. x A. y ( E. w ( y  e.  w  /\  w  e.  z )  ->  y  e.  x )  <->  E. x A. y ( E. x
( y  e.  x  /\  x  e.  z
)  ->  y  e.  x ) )
91, 8mpbi 208 1  |-  E. x A. y ( E. x
( y  e.  x  /\  x  e.  z
)  ->  y  e.  x )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1397   E.wex 1620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1621  df-nf 1625
This theorem is referenced by:  uniex2  6494  axunndlem1  8883
  Copyright terms: Public domain W3C validator