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Theorem elaltxp 25068
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 2872 . 2  |-  ( X  e.  ( A  XX.  B )  ->  X  e.  _V )
2 altopex 25053 . . . . 5  |-  << x ,  y >>  e.  _V
3 eleq1 2418 . . . . 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 2748 . 2  |-  ( E. x  e.  A  E. y  e.  B  X  =  << x ,  y
>>  ->  X  e.  _V )
7 eqeq1 2364 . . . 4  |-  ( z  =  X  ->  (
z  =  << x ,  y >> 
<->  X  =  << x ,  y >> ) )
872rexbidv 2662 . . 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 25052 . . 3  |-  ( A 
XX.  B )  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  << x ,  y
>> }
108, 9elab2g 2992 . 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 1642    e. wcel 1710   E.wrex 2620   _Vcvv 2864   <<caltop 25049    XX. caltxp 25050
This theorem is referenced by:  altopelaltxp  25069  altxpsspw  25070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-v 2866  df-dif 3231  df-un 3233  df-nul 3532  df-sn 3722  df-pr 3723  df-altop 25051  df-altxp 25052
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