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Theorem elaltxp 24509
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 2796 . 2  |-  ( X  e.  ( A  XX.  B )  ->  X  e.  _V )
2 altopex 24494 . . . . 5  |-  << x ,  y >>  e.  _V
3 eleq1 2343 . . . . 5  |-  ( X  =  << x ,  y
>>  ->  ( X  e. 
_V 
<-> 
<< x ,  y >>  e. 
_V ) )
42, 3mpbiri 224 . . . 4  |-  ( X  =  << x ,  y
>>  ->  X  e.  _V )
54a1i 10 . . 3  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( X  =  << x ,  y >>  ->  X  e.  _V ) )
65rexlimivv 2672 . 2  |-  ( E. x  e.  A  E. y  e.  B  X  =  << x ,  y
>>  ->  X  e.  _V )
7 eqeq1 2289 . . . 4  |-  ( z  =  X  ->  (
z  =  << x ,  y >> 
<->  X  =  << x ,  y >> ) )
872rexbidv 2586 . . 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 24493 . . 3  |-  ( A 
XX.  B )  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  << x ,  y
>> }
108, 9elab2g 2916 . 2  |-  ( X  e.  _V  ->  ( X  e.  ( A  XX. 
B )  <->  E. x  e.  A  E. y  e.  B  X  =  << x ,  y >> ) )
111, 6, 10pm5.21nii 342 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788   <<caltop 24490    XX. caltxp 24491
This theorem is referenced by:  altopelaltxp  24510  altxpsspw  24511
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-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-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-pr 3647  df-altop 24492  df-altxp 24493
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