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

Theorem axpr 4638
Description: Unabbreviated version of the Axiom of Pairing of ZF set theory, derived as a theorem from the other axioms.

This theorem should not be referenced by any proof. Instead, use ax-pr 4639 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.)

Assertion
Ref Expression
axpr  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
Distinct variable groups:    x, z, w    y, z, w

Proof of Theorem axpr
StepHypRef Expression
1 zfpair 4637 . . 3  |-  { x ,  y }  e.  _V
21isseti 3051 . 2  |-  E. z 
z  =  { x ,  y }
3 dfcleq 2445 . . 3  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
w  e.  { x ,  y } ) )
4 vex 3048 . . . . . . 7  |-  w  e. 
_V
54elpr 3986 . . . . . 6  |-  ( w  e.  { x ,  y }  <->  ( w  =  x  \/  w  =  y ) )
65bibi2i 315 . . . . 5  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  <->  ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
7 biimpr 202 . . . . 5  |-  ( ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) )  ->  ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
)
86, 7sylbi 199 . . . 4  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  ->  ( (
w  =  x  \/  w  =  y )  ->  w  e.  z ) )
98alimi 1684 . . 3  |-  ( A. w ( w  e.  z  <->  w  e.  { x ,  y } )  ->  A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z ) )
103, 9sylbi 199 . 2  |-  ( z  =  { x ,  y }  ->  A. w
( ( w  =  x  \/  w  =  y )  ->  w  e.  z ) )
112, 10eximii 1709 1  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370   A.wal 1442    = wceq 1444   E.wex 1663    e. wcel 1887   {cpr 3970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-pw 3953  df-sn 3969  df-pr 3971
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator