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Theorem altopelaltxp 29819
Description: Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 4960, 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 29818 . 2  |-  ( << X ,  Y >>  e.  ( A  XX.  B )  <->  E. x  e.  A  E. y  e.  B  << X ,  Y >>  =  << x ,  y >> )
2 reeanv 2967 . . 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 2405 . . . . 5  |-  ( << X ,  Y >>  =  << x ,  y >>  <->  << x ,  y >>  =  << X ,  Y >> )
4 vex 3054 . . . . . 6  |-  x  e. 
_V
5 vex 3054 . . . . . 6  |-  y  e. 
_V
64, 5altopth 29812 . . . . 5  |-  ( << x ,  y >>  =  << X ,  Y >>  <->  ( x  =  X  /\  y  =  Y ) )
73, 6bitri 249 . . . 4  |-  ( << X ,  Y >>  =  << x ,  y >>  <->  ( x  =  X  /\  y  =  Y ) )
872rexbii 2899 . . 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 2924 . . . 4  |-  ( X  e.  A  <->  E. x  e.  A  x  =  X )
10 risset 2924 . . . 4  |-  ( Y  e.  B  <->  E. y  e.  B  y  =  Y )
119, 10anbi12i 695 . . 3  |-  ( ( X  e.  A  /\  Y  e.  B )  <->  ( E. x  e.  A  x  =  X  /\  E. y  e.  B  y  =  Y ) )
122, 8, 113bitr4i 277 . 2  |-  ( E. x  e.  A  E. y  e.  B  << X ,  Y >>  =  << x ,  y >> 
<->  ( X  e.  A  /\  Y  e.  B
) )
131, 12bitri 249 1  |-  ( << X ,  Y >>  e.  ( A  XX.  B )  <->  ( X  e.  A  /\  Y  e.  B )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1836   E.wrex 2747   <<caltop 29799    XX. caltxp 29800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pr 4618
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-v 3053  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-sn 3962  df-pr 3964  df-altop 29801  df-altxp 29802
This theorem is referenced by: (None)
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