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Theorem altopelaltxp 24582
Description: Alternate ordered pair membership in a cross product. Note that, unlike opelxp 4735, there is no sethood requirement here. (Contributed by Scott Fenton, 22-Mar-2012.)
Assertion
Ref Expression
altopelaltxp  |-  ( << X ,  Y >>  e.  ( A  XX.  B )  <->  ( X  e.  A  /\  Y  e.  B )
)

Proof of Theorem altopelaltxp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elaltxp 24581 . 2  |-  ( << X ,  Y >>  e.  ( A  XX.  B )  <->  E. x  e.  A  E. y  e.  B  << X ,  Y >>  =  << x ,  y >> )
2 reeanv 2720 . . 3  |-  ( E. x  e.  A  E. y  e.  B  (
x  =  X  /\  y  =  Y )  <->  ( E. x  e.  A  x  =  X  /\  E. y  e.  B  y  =  Y ) )
3 eqcom 2298 . . . . 5  |-  ( << X ,  Y >>  =  << x ,  y >>  <->  << x ,  y >>  =  << X ,  Y >> )
4 vex 2804 . . . . . 6  |-  x  e. 
_V
5 vex 2804 . . . . . 6  |-  y  e. 
_V
64, 5altopth 24575 . . . . 5  |-  ( << x ,  y >>  =  << X ,  Y >>  <->  ( x  =  X  /\  y  =  Y ) )
73, 6bitri 240 . . . 4  |-  ( << X ,  Y >>  =  << x ,  y >>  <->  ( x  =  X  /\  y  =  Y ) )
872rexbii 2583 . . 3  |-  ( E. x  e.  A  E. y  e.  B  << X ,  Y >>  =  << x ,  y >> 
<->  E. x  e.  A  E. y  e.  B  ( x  =  X  /\  y  =  Y
) )
9 risset 2603 . . . 4  |-  ( X  e.  A  <->  E. x  e.  A  x  =  X )
10 risset 2603 . . . 4  |-  ( Y  e.  B  <->  E. y  e.  B  y  =  Y )
119, 10anbi12i 678 . . 3  |-  ( ( X  e.  A  /\  Y  e.  B )  <->  ( E. x  e.  A  x  =  X  /\  E. y  e.  B  y  =  Y ) )
122, 8, 113bitr4i 268 . 2  |-  ( E. x  e.  A  E. y  e.  B  << X ,  Y >>  =  << x ,  y >> 
<->  ( X  e.  A  /\  Y  e.  B
) )
131, 12bitri 240 1  |-  ( << X ,  Y >>  e.  ( A  XX.  B )  <->  ( X  e.  A  /\  Y  e.  B )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   <<caltop 24562    XX. caltxp 24563
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-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660  df-altop 24564  df-altxp 24565
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