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Theorem elaltxp 25812
Description: Membership in alternate cross 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 2956 . 2  |-  ( X  e.  ( A  XX.  B )  ->  X  e.  _V )
2 altopex 25797 . . . . 5  |-  << x ,  y >>  e.  _V
3 eleq1 2495 . . . . 5  |-  ( X  =  << x ,  y
>>  ->  ( X  e. 
_V 
<-> 
<< x ,  y >>  e. 
_V ) )
42, 3mpbiri 225 . . . 4  |-  ( X  =  << x ,  y
>>  ->  X  e.  _V )
54a1i 11 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( X  =  << x ,  y >>  ->  X  e.  _V ) )
65rexlimivv 2827 . 2  |-  ( E. x  e.  A  E. y  e.  B  X  =  << x ,  y
>>  ->  X  e.  _V )
7 eqeq1 2441 . . . 4  |-  ( z  =  X  ->  (
z  =  << x ,  y >> 
<->  X  =  << x ,  y >> ) )
872rexbidv 2740 . . 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 25796 . . 3  |-  ( A 
XX.  B )  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  << x ,  y
>> }
108, 9elab2g 3076 . 2  |-  ( X  e.  _V  ->  ( X  e.  ( A  XX. 
B )  <->  E. x  e.  A  E. y  e.  B  X  =  << x ,  y >> ) )
111, 6, 10pm5.21nii 343 1  |-  ( X  e.  ( A  XX.  B )  <->  E. x  e.  A  E. y  e.  B  X  =  << x ,  y >> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948   <<caltop 25793    XX. caltxp 25794
This theorem is referenced by:  altopelaltxp  25813  altxpsspw  25814
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-nul 3621  df-sn 3812  df-pr 3813  df-altop 25795  df-altxp 25796
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