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Theorem axpr 4675
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 4676 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 4674 . . 3  |-  { x ,  y }  e.  _V
21isseti 3112 . 2  |-  E. z 
z  =  { x ,  y }
3 dfcleq 2447 . . 3  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
w  e.  { x ,  y } ) )
4 vex 3109 . . . . . . 7  |-  w  e. 
_V
54elpr 4034 . . . . . 6  |-  ( w  e.  { x ,  y }  <->  ( w  =  x  \/  w  =  y ) )
65bibi2i 311 . . . . 5  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  <->  ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
7 bi2 198 . . . . 5  |-  ( ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) )  ->  ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
)
86, 7sylbi 195 . . . 4  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  ->  ( (
w  =  x  \/  w  =  y )  ->  w  e.  z ) )
98alimi 1638 . . 3  |-  ( A. w ( w  e.  z  <->  w  e.  { x ,  y } )  ->  A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z ) )
103, 9sylbi 195 . 2  |-  ( z  =  { x ,  y }  ->  A. w
( ( w  =  x  \/  w  =  y )  ->  w  e.  z ) )
112, 10eximii 1663 1  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366   A.wal 1396    = wceq 1398   E.wex 1617    e. wcel 1823   {cpr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-pw 4001  df-sn 4017  df-pr 4019
This theorem is referenced by: (None)
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