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Theorem axpr 4151
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 4152 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 4150 . . 3  |-  { x ,  y }  e.  _V
21isseti 2746 . 2  |-  E. z 
z  =  { x ,  y }
3 dfcleq 2250 . . . 4  |-  ( z  =  { x ,  y }  <->  A. w
( w  e.  z  <-> 
w  e.  { x ,  y } ) )
4 vex 2743 . . . . . . . 8  |-  w  e. 
_V
54elpr 3599 . . . . . . 7  |-  ( w  e.  { x ,  y }  <->  ( w  =  x  \/  w  =  y ) )
65bibi2i 306 . . . . . 6  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  <->  ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) ) )
7 bi2 191 . . . . . 6  |-  ( ( w  e.  z  <->  ( w  =  x  \/  w  =  y ) )  ->  ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
)
86, 7sylbi 189 . . . . 5  |-  ( ( w  e.  z  <->  w  e.  { x ,  y } )  ->  ( (
w  =  x  \/  w  =  y )  ->  w  e.  z ) )
98alimi 1546 . . . 4  |-  ( A. w ( w  e.  z  <->  w  e.  { x ,  y } )  ->  A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z ) )
103, 9sylbi 189 . . 3  |-  ( z  =  { x ,  y }  ->  A. w
( ( w  =  x  \/  w  =  y )  ->  w  e.  z ) )
1110eximi 1574 . 2  |-  ( E. z  z  =  {
x ,  y }  ->  E. z A. w
( ( w  =  x  \/  w  =  y )  ->  w  e.  z ) )
122, 11ax-mp 10 1  |-  E. z A. w ( ( w  =  x  \/  w  =  y )  ->  w  e.  z )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359   A.wal 1532   E.wex 1537    = wceq 1619    e. wcel 1621   {cpr 3582
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-pw 3568  df-sn 3587  df-pr 3588
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