Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elaltxp Structured version   Unicode version

Theorem elaltxp 30326
Description: Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.)
Assertion
Ref Expression
elaltxp  |-  ( X  e.  ( A  XX.  B )  <->  E. x  e.  A  E. y  e.  B  X  =  << x ,  y >> )
Distinct variable groups:    x, A, y    x, B, y    x, X, y

Proof of Theorem elaltxp
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 elex 3070 . 2  |-  ( X  e.  ( A  XX.  B )  ->  X  e.  _V )
2 altopex 30311 . . . . 5  |-  << x ,  y >>  e.  _V
3 eleq1 2476 . . . . 5  |-  ( X  =  << x ,  y
>>  ->  ( X  e. 
_V 
<-> 
<< x ,  y >>  e. 
_V ) )
42, 3mpbiri 235 . . . 4  |-  ( X  =  << x ,  y
>>  ->  X  e.  _V )
54a1i 11 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( X  =  << x ,  y >>  ->  X  e.  _V ) )
65rexlimivv 2903 . 2  |-  ( E. x  e.  A  E. y  e.  B  X  =  << x ,  y
>>  ->  X  e.  _V )
7 eqeq1 2408 . . . 4  |-  ( z  =  X  ->  (
z  =  << x ,  y >> 
<->  X  =  << x ,  y >> ) )
872rexbidv 2927 . . 3  |-  ( z  =  X  ->  ( E. x  e.  A  E. y  e.  B  z  =  << x ,  y >> 
<->  E. x  e.  A  E. y  e.  B  X  =  << x ,  y >> ) )
9 df-altxp 30310 . . 3  |-  ( A 
XX.  B )  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  << x ,  y
>> }
108, 9elab2g 3200 . 2  |-  ( X  e.  _V  ->  ( X  e.  ( A  XX. 
B )  <->  E. x  e.  A  E. y  e.  B  X  =  << x ,  y >> ) )
111, 6, 10pm5.21nii 353 1  |-  ( X  e.  ( A  XX.  B )  <->  E. x  e.  A  E. y  e.  B  X  =  << x ,  y >> )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   E.wrex 2757   _Vcvv 3061   <<caltop 30307    XX. caltxp 30308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-v 3063  df-dif 3419  df-un 3421  df-nul 3741  df-sn 3975  df-pr 3977  df-altop 30309  df-altxp 30310
This theorem is referenced by:  altopelaltxp  30327  altxpsspw  30328
  Copyright terms: Public domain W3C validator