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Theorem elxp7 6832
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 6743. (Contributed by NM, 19-Aug-2006.)
Assertion
Ref Expression
elxp7  |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A
)  e.  C ) ) )

Proof of Theorem elxp7
StepHypRef Expression
1 elxp6 6831 . 2  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
2 fvex 5882 . . . . 5  |-  ( 1st `  A )  e.  _V
3 fvex 5882 . . . . 5  |-  ( 2nd `  A )  e.  _V
42, 3pm3.2i 455 . . . 4  |-  ( ( 1st `  A )  e.  _V  /\  ( 2nd `  A )  e. 
_V )
5 elxp6 6831 . . . 4  |-  ( A  e.  ( _V  X.  _V )  <->  ( A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) ) )
64, 5mpbiran2 919 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
76anbi1i 695 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  <-> 
( A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) ) )
81, 7bitr4i 252 1  |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A
)  e.  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   <.cop 4038    X. cxp 5006   ` cfv 5594   1stc1st 6797   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fv 5602  df-1st 6799  df-2nd 6800
This theorem is referenced by:  xp2  6834  unielxp  6835  1stconst  6887  2ndconst  6888  fparlem1  6899  fparlem2  6900  infxpenlem  8408  1stpreimas  27679  1stpreima  27680  2ndpreima  27681  f1od2  27704  xpinpreima2  28050  tpr2rico  28055  sxbrsigalem0  28415  dya2iocnrect  28425  pellex  30975
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