MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  zfpair2 Structured version   Unicode version

Theorem zfpair2 4677
Description: Derive the abbreviated version of the Axiom of Pairing from ax-pr 4676. See zfpair 4674 for its derivation from the other axioms. (Contributed by NM, 14-Nov-2006.)
Assertion
Ref Expression
zfpair2  |-  { x ,  y }  e.  _V

Proof of Theorem zfpair2
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-pr 4676 . . . 4  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
21bm1.3ii 4561 . . 3  |-  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) )
3 dfcleq 2436 . . . . 5  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
w  e.  { x ,  y } ) )
4 vex 3098 . . . . . . . 8  |-  w  e. 
_V
54elpr 4032 . . . . . . 7  |-  ( w  e.  { x ,  y }  <->  ( w  =  x  \/  w  =  y ) )
65bibi2i 313 . . . . . 6  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  <->  ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
76albii 1627 . . . . 5  |-  ( A. w ( w  e.  z  <->  w  e.  { x ,  y } )  <->  A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
83, 7bitri 249 . . . 4  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
( w  =  x  \/  w  =  y ) ) )
98exbii 1654 . . 3  |-  ( E. z  z  =  {
x ,  y }  <->  E. z A. w ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
102, 9mpbir 209 . 2  |-  E. z 
z  =  { x ,  y }
1110issetri 3102 1  |-  { x ,  y }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368   A.wal 1381    = wceq 1383   E.wex 1599    e. wcel 1804   _Vcvv 3095   {cpr 4016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-v 3097  df-un 3466  df-sn 4015  df-pr 4017
This theorem is referenced by:  snex  4678  prex  4679  pwssun  4776  xpsspwOLD  5107  funopg  5610  fiint  7799  brdom7disj  8912  brdom6disj  8913  wlkntrllem1  24539  frisusgranb  24975  2pthfrgrarn  24987
  Copyright terms: Public domain W3C validator