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Theorem zfpair 4150
Description: The Axiom of Pairing of Zermelo-Fraenkel set theory. Axiom 2 of [TakeutiZaring] p. 15. In some textbooks this is stated as a separate axiom; here we show it is redundant since it can be derived from the other axioms.

This theorem should not be referenced by any proof other than axpr 4151. Instead, use zfpair2 4153 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 18-Oct-1995.) (New usage is discouraged.)

Assertion
Ref Expression
zfpair  |-  { x ,  y }  e.  _V

Proof of Theorem zfpair
StepHypRef Expression
1 dfpr2 3597 . 2  |-  { x ,  y }  =  { w  |  (
w  =  x  \/  w  =  y ) }
2 19.43 1604 . . . . 5  |-  ( E. z ( ( z  =  (/)  /\  w  =  x )  \/  (
z  =  { (/) }  /\  w  =  y ) )  <->  ( E. z ( z  =  (/)  /\  w  =  x )  \/  E. z
( z  =  { (/)
}  /\  w  =  y ) ) )
3 prlem2 934 . . . . . 6  |-  ( ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y ) )  <-> 
( ( z  =  (/)  \/  z  =  { (/)
} )  /\  (
( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y ) ) ) )
43exbii 1580 . . . . 5  |-  ( E. z ( ( z  =  (/)  /\  w  =  x )  \/  (
z  =  { (/) }  /\  w  =  y ) )  <->  E. z
( ( z  =  (/)  \/  z  =  { (/)
} )  /\  (
( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y ) ) ) )
5 0ex 4090 . . . . . . . 8  |-  (/)  e.  _V
65isseti 2746 . . . . . . 7  |-  E. z 
z  =  (/)
7 19.41v 2035 . . . . . . 7  |-  ( E. z ( z  =  (/)  /\  w  =  x )  <->  ( E. z 
z  =  (/)  /\  w  =  x ) )
86, 7mpbiran 889 . . . . . 6  |-  ( E. z ( z  =  (/)  /\  w  =  x )  <->  w  =  x
)
9 p0ex 4135 . . . . . . . 8  |-  { (/) }  e.  _V
109isseti 2746 . . . . . . 7  |-  E. z 
z  =  { (/) }
11 19.41v 2035 . . . . . . 7  |-  ( E. z ( z  =  { (/) }  /\  w  =  y )  <->  ( E. z  z  =  { (/)
}  /\  w  =  y ) )
1210, 11mpbiran 889 . . . . . 6  |-  ( E. z ( z  =  { (/) }  /\  w  =  y )  <->  w  =  y )
138, 12orbi12i 509 . . . . 5  |-  ( ( E. z ( z  =  (/)  /\  w  =  x )  \/  E. z ( z  =  { (/) }  /\  w  =  y ) )  <-> 
( w  =  x  \/  w  =  y ) )
142, 4, 133bitr3ri 269 . . . 4  |-  ( ( w  =  x  \/  w  =  y )  <->  E. z ( ( z  =  (/)  \/  z  =  { (/) } )  /\  ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y )
) ) )
1514abbii 2368 . . 3  |-  { w  |  ( w  =  x  \/  w  =  y ) }  =  { w  |  E. z ( ( z  =  (/)  \/  z  =  { (/) } )  /\  ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y )
) ) }
16 dfpr2 3597 . . . . 5  |-  { (/) ,  { (/) } }  =  { z  |  ( z  =  (/)  \/  z  =  { (/) } ) }
17 pp0ex 4137 . . . . 5  |-  { (/) ,  { (/) } }  e.  _V
1816, 17eqeltrri 2327 . . . 4  |-  { z  |  ( z  =  (/)  \/  z  =  { (/)
} ) }  e.  _V
19 equequ2 1830 . . . . . . . 8  |-  ( v  =  x  ->  (
w  =  v  <->  w  =  x ) )
20 0inp0 4120 . . . . . . . 8  |-  ( z  =  (/)  ->  -.  z  =  { (/) } )
2119, 20prlem1 933 . . . . . . 7  |-  ( v  =  x  ->  (
z  =  (/)  ->  (
( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y )
)  ->  w  =  v ) ) )
2221alrimdv 2015 . . . . . 6  |-  ( v  =  x  ->  (
z  =  (/)  ->  A. w
( ( ( z  =  (/)  /\  w  =  x )  \/  (
z  =  { (/) }  /\  w  =  y ) )  ->  w  =  v ) ) )
2322a4imev 1998 . . . . 5  |-  ( z  =  (/)  ->  E. v A. w ( ( ( z  =  (/)  /\  w  =  x )  \/  (
z  =  { (/) }  /\  w  =  y ) )  ->  w  =  v ) )
24 orcom 378 . . . . . . . 8  |-  ( ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y ) )  <-> 
( ( z  =  { (/) }  /\  w  =  y )  \/  ( z  =  (/)  /\  w  =  x ) ) )
25 equequ2 1830 . . . . . . . . 9  |-  ( v  =  y  ->  (
w  =  v  <->  w  =  y ) )
2620con2i 114 . . . . . . . . 9  |-  ( z  =  { (/) }  ->  -.  z  =  (/) )
2725, 26prlem1 933 . . . . . . . 8  |-  ( v  =  y  ->  (
z  =  { (/) }  ->  ( ( ( z  =  { (/) }  /\  w  =  y )  \/  ( z  =  (/)  /\  w  =  x ) )  ->  w  =  v )
) )
2824, 27syl7bi 223 . . . . . . 7  |-  ( v  =  y  ->  (
z  =  { (/) }  ->  ( ( ( z  =  (/)  /\  w  =  x )  \/  (
z  =  { (/) }  /\  w  =  y ) )  ->  w  =  v ) ) )
2928alrimdv 2015 . . . . . 6  |-  ( v  =  y  ->  (
z  =  { (/) }  ->  A. w ( ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y ) )  ->  w  =  v ) ) )
3029a4imev 1998 . . . . 5  |-  ( z  =  { (/) }  ->  E. v A. w ( ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y )
)  ->  w  =  v ) )
3123, 30jaoi 370 . . . 4  |-  ( ( z  =  (/)  \/  z  =  { (/) } )  ->  E. v A. w ( ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y )
)  ->  w  =  v ) )
3218, 31zfrep4 4079 . . 3  |-  { w  |  E. z ( ( z  =  (/)  \/  z  =  { (/) } )  /\  ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y )
) ) }  e.  _V
3315, 32eqeltri 2326 . 2  |-  { w  |  ( w  =  x  \/  w  =  y ) }  e.  _V
341, 33eqeltri 2326 1  |-  { x ,  y }  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360   A.wal 1532   E.wex 1537    = wceq 1619    e. wcel 1621   {cab 2242   _Vcvv 2740   (/)c0 3397   {csn 3581   {cpr 3582
This theorem is referenced by:  axpr  4151  isdrs2  14000  clatl  14147  dfdir2  24623  latdir  24627
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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