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Theorem zfpair 4401
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 4402. Instead, use zfpair2 4404 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
Dummy variables  z  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfpr2 3830 . 2  |-  { x ,  y }  =  { w  |  (
w  =  x  \/  w  =  y ) }
2 19.43 1615 . . . . 5  |-  ( E. z ( ( z  =  (/)  /\  w  =  x )  \/  (
z  =  { (/) }  /\  w  =  y ) )  <->  ( E. z ( z  =  (/)  /\  w  =  x )  \/  E. z
( z  =  { (/)
}  /\  w  =  y ) ) )
3 prlem2 930 . . . . . 6  |-  ( ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y ) )  <-> 
( ( z  =  (/)  \/  z  =  { (/)
} )  /\  (
( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y ) ) ) )
43exbii 1592 . . . . 5  |-  ( E. z ( ( z  =  (/)  /\  w  =  x )  \/  (
z  =  { (/) }  /\  w  =  y ) )  <->  E. z
( ( z  =  (/)  \/  z  =  { (/)
} )  /\  (
( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y ) ) ) )
5 0ex 4339 . . . . . . . 8  |-  (/)  e.  _V
65isseti 2962 . . . . . . 7  |-  E. z 
z  =  (/)
7 19.41v 1924 . . . . . . 7  |-  ( E. z ( z  =  (/)  /\  w  =  x )  <->  ( E. z 
z  =  (/)  /\  w  =  x ) )
86, 7mpbiran 885 . . . . . 6  |-  ( E. z ( z  =  (/)  /\  w  =  x )  <->  w  =  x
)
9 p0ex 4386 . . . . . . . 8  |-  { (/) }  e.  _V
109isseti 2962 . . . . . . 7  |-  E. z 
z  =  { (/) }
11 19.41v 1924 . . . . . . 7  |-  ( E. z ( z  =  { (/) }  /\  w  =  y )  <->  ( E. z  z  =  { (/)
}  /\  w  =  y ) )
1210, 11mpbiran 885 . . . . . 6  |-  ( E. z ( z  =  { (/) }  /\  w  =  y )  <->  w  =  y )
138, 12orbi12i 508 . . . . 5  |-  ( ( E. z ( z  =  (/)  /\  w  =  x )  \/  E. z ( z  =  { (/) }  /\  w  =  y ) )  <-> 
( w  =  x  \/  w  =  y ) )
142, 4, 133bitr3ri 268 . . . 4  |-  ( ( w  =  x  \/  w  =  y )  <->  E. z ( ( z  =  (/)  \/  z  =  { (/) } )  /\  ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y )
) ) )
1514abbii 2548 . . 3  |-  { w  |  ( w  =  x  \/  w  =  y ) }  =  { w  |  E. z ( ( z  =  (/)  \/  z  =  { (/) } )  /\  ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y )
) ) }
16 dfpr2 3830 . . . . 5  |-  { (/) ,  { (/) } }  =  { z  |  ( z  =  (/)  \/  z  =  { (/) } ) }
17 pp0ex 4388 . . . . 5  |-  { (/) ,  { (/) } }  e.  _V
1816, 17eqeltrri 2507 . . . 4  |-  { z  |  ( z  =  (/)  \/  z  =  { (/)
} ) }  e.  _V
19 equequ2 1698 . . . . . . . 8  |-  ( v  =  x  ->  (
w  =  v  <->  w  =  x ) )
20 0inp0 4371 . . . . . . . 8  |-  ( z  =  (/)  ->  -.  z  =  { (/) } )
2119, 20prlem1 929 . . . . . . 7  |-  ( v  =  x  ->  (
z  =  (/)  ->  (
( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y )
)  ->  w  =  v ) ) )
2221alrimdv 1643 . . . . . 6  |-  ( v  =  x  ->  (
z  =  (/)  ->  A. w
( ( ( z  =  (/)  /\  w  =  x )  \/  (
z  =  { (/) }  /\  w  =  y ) )  ->  w  =  v ) ) )
2322spimev 1964 . . . . 5  |-  ( z  =  (/)  ->  E. v A. w ( ( ( z  =  (/)  /\  w  =  x )  \/  (
z  =  { (/) }  /\  w  =  y ) )  ->  w  =  v ) )
24 orcom 377 . . . . . . . 8  |-  ( ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y ) )  <-> 
( ( z  =  { (/) }  /\  w  =  y )  \/  ( z  =  (/)  /\  w  =  x ) ) )
25 equequ2 1698 . . . . . . . . 9  |-  ( v  =  y  ->  (
w  =  v  <->  w  =  y ) )
2620con2i 114 . . . . . . . . 9  |-  ( z  =  { (/) }  ->  -.  z  =  (/) )
2725, 26prlem1 929 . . . . . . . 8  |-  ( v  =  y  ->  (
z  =  { (/) }  ->  ( ( ( z  =  { (/) }  /\  w  =  y )  \/  ( z  =  (/)  /\  w  =  x ) )  ->  w  =  v )
) )
2824, 27syl7bi 222 . . . . . . 7  |-  ( v  =  y  ->  (
z  =  { (/) }  ->  ( ( ( z  =  (/)  /\  w  =  x )  \/  (
z  =  { (/) }  /\  w  =  y ) )  ->  w  =  v ) ) )
2928alrimdv 1643 . . . . . 6  |-  ( v  =  y  ->  (
z  =  { (/) }  ->  A. w ( ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y ) )  ->  w  =  v ) ) )
3029spimev 1964 . . . . 5  |-  ( z  =  { (/) }  ->  E. v A. w ( ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y )
)  ->  w  =  v ) )
3123, 30jaoi 369 . . . 4  |-  ( ( z  =  (/)  \/  z  =  { (/) } )  ->  E. v A. w ( ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y )
)  ->  w  =  v ) )
3218, 31zfrep4 4328 . . 3  |-  { w  |  E. z ( ( z  =  (/)  \/  z  =  { (/) } )  /\  ( ( z  =  (/)  /\  w  =  x )  \/  ( z  =  { (/) }  /\  w  =  y )
) ) }  e.  _V
3315, 32eqeltri 2506 . 2  |-  { w  |  ( w  =  x  \/  w  =  y ) }  e.  _V
341, 33eqeltri 2506 1  |-  { x ,  y }  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422   _Vcvv 2956   (/)c0 3628   {csn 3814   {cpr 3815
This theorem is referenced by:  axpr  4402  clatl  14543
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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