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Theorem elvvv 4917
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 4876 . 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 649 . . . . 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 1926 . . . . . 6  |-  ( E. x E. y ( A  =  <. w ,  z >.  /\  w  =  <. x ,  y
>. )  <->  ( A  = 
<. w ,  z >.  /\  E. x E. y  w  =  <. x ,  y >. ) )
4 ancom 450 . . . . . . 7  |-  ( ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. )  <->  ( A  = 
<. w ,  z >.  /\  w  =  <. x ,  y >. )
)
542exbii 1635 . . . . . 6  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. )  <->  E. x E. y
( A  =  <. w ,  z >.  /\  w  =  <. x ,  y
>. ) )
6 vex 2994 . . . . . . . 8  |-  z  e. 
_V
76biantru 505 . . . . . . 7  |-  ( ( A  =  <. w ,  z >.  /\  w  e.  ( _V  X.  _V ) )  <->  ( ( A  =  <. w ,  z >.  /\  w  e.  ( _V  X.  _V ) )  /\  z  e.  _V ) )
8 elvv 4916 . . . . . . . 8  |-  ( w  e.  ( _V  X.  _V )  <->  E. x E. y  w  =  <. x ,  y >. )
98anbi2i 694 . . . . . . 7  |-  ( ( A  =  <. w ,  z >.  /\  w  e.  ( _V  X.  _V ) )  <->  ( A  =  <. w ,  z
>.  /\  E. x E. y  w  =  <. x ,  y >. )
)
107, 9bitr3i 251 . . . . . 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 278 . . . . 5  |-  ( ( ( A  =  <. w ,  z >.  /\  w  e.  ( _V  X.  _V ) )  /\  z  e.  _V )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  A  =  <. w ,  z >. )
)
122, 11bitr3i 251 . . . 4  |-  ( ( A  =  <. w ,  z >.  /\  (
w  e.  ( _V 
X.  _V )  /\  z  e.  _V ) )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  A  =  <. w ,  z >. )
)
13122exbii 1635 . . 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 1791 . . . 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 1787 . . . . . 6  |-  ( E. w E. z ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. )  <->  E. z E. w
( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. ) )
16 opex 4575 . . . . . . . 8  |-  <. x ,  y >.  e.  _V
17 opeq1 4078 . . . . . . . . 9  |-  ( w  =  <. x ,  y
>.  ->  <. w ,  z
>.  =  <. <. x ,  y >. ,  z
>. )
1817eqeq2d 2454 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( A  = 
<. w ,  z >.  <->  A  =  <. <. x ,  y
>. ,  z >. ) )
1916, 18ceqsexv 3028 . . . . . . 7  |-  ( E. w ( w  = 
<. x ,  y >.  /\  A  =  <. w ,  z >. )  <->  A  =  <. <. x ,  y
>. ,  z >. )
2019exbii 1634 . . . . . 6  |-  ( E. z E. w ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. )  <->  E. z  A  = 
<. <. x ,  y
>. ,  z >. )
2115, 20bitri 249 . . . . 5  |-  ( E. w E. z ( w  =  <. x ,  y >.  /\  A  =  <. w ,  z
>. )  <->  E. z  A  = 
<. <. x ,  y
>. ,  z >. )
22212exbii 1635 . . . 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 251 . . 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 249 . 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 249 1  |-  ( A  e.  ( ( _V 
X.  _V )  X.  _V ) 
<->  E. x E. y E. z  A  =  <. <. x ,  y
>. ,  z >. )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2991   <.cop 3902    X. cxp 4857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-rab 2743  df-v 2993  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-opab 4370  df-xp 4865
This theorem is referenced by:  ssrelrel  4959  dftpos3  6782
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