MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axext2 Unicode version

Theorem axext2 2278
Description: The Axiom of Extensionality (ax-ext 2277) restated so that it postulates the existence of a set  z given two arbitrary sets 
x and  y. This way to express it follows the general idea of the other ZFC axioms, which is to postulate the existence of sets given other sets. (Contributed by NM, 28-Sep-2003.)
Assertion
Ref Expression
axext2  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
Distinct variable group:    x, y, z

Proof of Theorem axext2
StepHypRef Expression
1 ax-ext 2277 . 2  |-  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
2 19.36v 1849 . 2  |-  ( E. z ( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )  <->  ( A. z ( z  e.  x  <->  z  e.  y )  ->  x  =  y ) )
31, 2mpbir 200 1  |-  E. z
( ( z  e.  x  <->  z  e.  y )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696
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-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535
  Copyright terms: Public domain W3C validator