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Theorem elaltxp 24581
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 2809 . 2  |-  ( X  e.  ( A  XX.  B )  ->  X  e.  _V )
2 altopex 24566 . . . . 5  |-  << x ,  y >>  e.  _V
3 eleq1 2356 . . . . 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 2685 . 2  |-  ( E. x  e.  A  E. y  e.  B  X  =  << x ,  y
>>  ->  X  e.  _V )
7 eqeq1 2302 . . . 4  |-  ( z  =  X  ->  (
z  =  << x ,  y >> 
<->  X  =  << x ,  y >> ) )
872rexbidv 2599 . . 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 24565 . . 3  |-  ( A 
XX.  B )  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  << x ,  y
>> }
108, 9elab2g 2929 . 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 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801   <<caltop 24562    XX. caltxp 24563
This theorem is referenced by:  altopelaltxp  24582  altxpsspw  24583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-sn 3659  df-pr 3660  df-altop 24564  df-altxp 24565
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