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Theorem altopelaltxp 28171
Description: Alternate ordered pair membership in a Cartesian product. Note that, unlike opelxp 4980, 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 28170 . 2  |-  ( << X ,  Y >>  e.  ( A  XX.  B )  <->  E. x  e.  A  E. y  e.  B  << X ,  Y >>  =  << x ,  y >> )
2 reeanv 2994 . . 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 2463 . . . . 5  |-  ( << X ,  Y >>  =  << x ,  y >>  <->  << x ,  y >>  =  << X ,  Y >> )
4 vex 3081 . . . . . 6  |-  x  e. 
_V
5 vex 3081 . . . . . 6  |-  y  e. 
_V
64, 5altopth 28164 . . . . 5  |-  ( << x ,  y >>  =  << X ,  Y >>  <->  ( x  =  X  /\  y  =  Y ) )
73, 6bitri 249 . . . 4  |-  ( << X ,  Y >>  =  << x ,  y >>  <->  ( x  =  X  /\  y  =  Y ) )
872rexbii 2863 . . 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 2885 . . . 4  |-  ( X  e.  A  <->  E. x  e.  A  x  =  X )
10 risset 2885 . . . 4  |-  ( Y  e.  B  <->  E. y  e.  B  y  =  Y )
119, 10anbi12i 697 . . 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 369    = wceq 1370    e. wcel 1758   E.wrex 2800   <<caltop 28151    XX. caltxp 28152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-sn 3989  df-pr 3991  df-altop 28153  df-altxp 28154
This theorem is referenced by: (None)
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