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Theorem axpr 4402
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 4403 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 4401 . . 3  |-  { x ,  y }  e.  _V
21isseti 2962 . 2  |-  E. z 
z  =  { x ,  y }
3 dfcleq 2430 . . 3  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
w  e.  { x ,  y } ) )
4 vex 2959 . . . . . . 7  |-  w  e. 
_V
54elpr 3832 . . . . . 6  |-  ( w  e.  { x ,  y }  <->  ( w  =  x  \/  w  =  y ) )
65bibi2i 305 . . . . 5  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  <->  ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
7 bi2 190 . . . . 5  |-  ( ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) )  ->  ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
)
86, 7sylbi 188 . . . 4  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  ->  ( (
w  =  x  \/  w  =  y )  ->  w  e.  z ) )
98alimi 1568 . . 3  |-  ( A. w ( w  e.  z  <->  w  e.  { x ,  y } )  ->  A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z ) )
103, 9sylbi 188 . 2  |-  ( z  =  { x ,  y }  ->  A. w
( ( w  =  x  \/  w  =  y )  ->  w  e.  z ) )
112, 10eximii 1587 1  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   {cpr 3815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-pw 3801  df-sn 3820  df-pr 3821
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