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Theorem elvvv 4749
Description: Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)
Assertion
Ref Expression
elvvv  |-  ( A  e.  ( ( _V 
X.  _V )  X.  _V ) 
<->  E. x E. y E. z  A  =  <. <. x ,  y
>. ,  z >. )
Distinct variable group:    x, y, z, A

Proof of Theorem elvvv
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elxp 4706 . 2  |-  ( A  e.  ( ( _V 
X.  _V )  X.  _V ) 
<->  E. w E. z
( A  =  <. w ,  z >.  /\  (
w  e.  ( _V 
X.  _V )  /\  z  e.  _V ) ) )
2 anass 630 . . . . 5  |-  ( ( ( A  =  <. w ,  z >.  /\  w  e.  ( _V  X.  _V ) )  /\  z  e.  _V )  <->  ( A  =  <. w ,  z
>.  /\  ( w  e.  ( _V  X.  _V )  /\  z  e.  _V ) ) )
3 19.42vv 1848 . . . . . 6  |-  ( E. x E. y ( A  =  <. w ,  z >.  /\  w  =  <. x ,  y
>. )  <->  ( A  = 
<. w ,  z >.  /\  E. x E. y  w  =  <. x ,  y >. ) )
4 ancom 437 . . . . . . 7  |-  ( ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. )  <->  ( A  = 
<. w ,  z >.  /\  w  =  <. x ,  y >. )
)
542exbii 1570 . . . . . 6  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. )  <->  E. x E. y
( A  =  <. w ,  z >.  /\  w  =  <. x ,  y
>. ) )
6 vex 2791 . . . . . . . 8  |-  z  e. 
_V
76biantru 491 . . . . . . 7  |-  ( ( A  =  <. w ,  z >.  /\  w  e.  ( _V  X.  _V ) )  <->  ( ( A  =  <. w ,  z >.  /\  w  e.  ( _V  X.  _V ) )  /\  z  e.  _V ) )
8 elvv 4748 . . . . . . . 8  |-  ( w  e.  ( _V  X.  _V )  <->  E. x E. y  w  =  <. x ,  y >. )
98anbi2i 675 . . . . . . 7  |-  ( ( A  =  <. w ,  z >.  /\  w  e.  ( _V  X.  _V ) )  <->  ( A  =  <. w ,  z
>.  /\  E. x E. y  w  =  <. x ,  y >. )
)
107, 9bitr3i 242 . . . . . 6  |-  ( ( ( A  =  <. w ,  z >.  /\  w  e.  ( _V  X.  _V ) )  /\  z  e.  _V )  <->  ( A  =  <. w ,  z
>.  /\  E. x E. y  w  =  <. x ,  y >. )
)
113, 5, 103bitr4ri 269 . . . . 5  |-  ( ( ( A  =  <. w ,  z >.  /\  w  e.  ( _V  X.  _V ) )  /\  z  e.  _V )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  A  =  <. w ,  z >. )
)
122, 11bitr3i 242 . . . 4  |-  ( ( A  =  <. w ,  z >.  /\  (
w  e.  ( _V 
X.  _V )  /\  z  e.  _V ) )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  A  =  <. w ,  z >. )
)
13122exbii 1570 . . 3  |-  ( E. w E. z ( A  =  <. w ,  z >.  /\  (
w  e.  ( _V 
X.  _V )  /\  z  e.  _V ) )  <->  E. w E. z E. x E. y ( w  = 
<. x ,  y >.  /\  A  =  <. w ,  z >. )
)
14 exrot4 1819 . . . 4  |-  ( E. x E. y E. w E. z ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. )  <->  E. w E. z E. x E. y ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. ) )
15 excom 1786 . . . . . 6  |-  ( E. w E. z ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. )  <->  E. z E. w
( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. ) )
16 opex 4237 . . . . . . . 8  |-  <. x ,  y >.  e.  _V
17 opeq1 3796 . . . . . . . . 9  |-  ( w  =  <. x ,  y
>.  ->  <. w ,  z
>.  =  <. <. x ,  y >. ,  z
>. )
1817eqeq2d 2294 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( A  = 
<. w ,  z >.  <->  A  =  <. <. x ,  y
>. ,  z >. ) )
1916, 18ceqsexv 2823 . . . . . . 7  |-  ( E. w ( w  = 
<. x ,  y >.  /\  A  =  <. w ,  z >. )  <->  A  =  <. <. x ,  y
>. ,  z >. )
2019exbii 1569 . . . . . 6  |-  ( E. z E. w ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. )  <->  E. z  A  = 
<. <. x ,  y
>. ,  z >. )
2115, 20bitri 240 . . . . 5  |-  ( E. w E. z ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. )  <->  E. z  A  = 
<. <. x ,  y
>. ,  z >. )
22212exbii 1570 . . . 4  |-  ( E. x E. y E. w E. z ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. )  <->  E. x E. y E. z  A  =  <. <. x ,  y
>. ,  z >. )
2314, 22bitr3i 242 . . 3  |-  ( E. w E. z E. x E. y ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. )  <->  E. x E. y E. z  A  =  <. <. x ,  y
>. ,  z >. )
2413, 23bitri 240 . 2  |-  ( E. w E. z ( A  =  <. w ,  z >.  /\  (
w  e.  ( _V 
X.  _V )  /\  z  e.  _V ) )  <->  E. x E. y E. z  A  =  <. <. x ,  y
>. ,  z >. )
251, 24bitri 240 1  |-  ( A  e.  ( ( _V 
X.  _V )  X.  _V ) 
<->  E. x E. y E. z  A  =  <. <. x ,  y
>. ,  z >. )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    X. cxp 4687
This theorem is referenced by:  ssrelrel  4787  dftpos3  6252
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695
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