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Theorem altopelaltxp 25786
Description: Alternate ordered pair membership in a cross product. Note that, unlike opelxp 4900, 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 25785 . 2  |-  ( << X ,  Y >>  e.  ( A  XX.  B )  <->  E. x  e.  A  E. y  e.  B  << X ,  Y >>  =  << x ,  y >> )
2 reeanv 2867 . . 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 2437 . . . . 5  |-  ( << X ,  Y >>  =  << x ,  y >>  <->  << x ,  y >>  =  << X ,  Y >> )
4 vex 2951 . . . . . 6  |-  x  e. 
_V
5 vex 2951 . . . . . 6  |-  y  e. 
_V
64, 5altopth 25779 . . . . 5  |-  ( << x ,  y >>  =  << X ,  Y >>  <->  ( x  =  X  /\  y  =  Y ) )
73, 6bitri 241 . . . 4  |-  ( << X ,  Y >>  =  << x ,  y >>  <->  ( x  =  X  /\  y  =  Y ) )
872rexbii 2724 . . 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 2745 . . . 4  |-  ( X  e.  A  <->  E. x  e.  A  x  =  X )
10 risset 2745 . . . 4  |-  ( Y  e.  B  <->  E. y  e.  B  y  =  Y )
119, 10anbi12i 679 . . 3  |-  ( ( X  e.  A  /\  Y  e.  B )  <->  ( E. x  e.  A  x  =  X  /\  E. y  e.  B  y  =  Y ) )
122, 8, 113bitr4i 269 . 2  |-  ( E. x  e.  A  E. y  e.  B  << X ,  Y >>  =  << x ,  y >> 
<->  ( X  e.  A  /\  Y  e.  B
) )
131, 12bitri 241 1  |-  ( << X ,  Y >>  e.  ( A  XX.  B )  <->  ( X  e.  A  /\  Y  e.  B )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   <<caltop 25766    XX. caltxp 25767
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-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-pr 3813  df-altop 25768  df-altxp 25769
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