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Theorem elxp7 5979
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4952. (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 elex 2644 . 2  |-  ( A  e.  ( B  X.  C )  ->  A  e.  _V )
2 elex 2644 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  A  e. 
_V )
32adantr 271 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  A  e.  _V )
4 1stexg 5976 . . . . . . 7  |-  ( A  e.  _V  ->  ( 1st `  A )  e. 
_V )
5 2ndexg 5977 . . . . . . 7  |-  ( A  e.  _V  ->  ( 2nd `  A )  e. 
_V )
64, 5jca 301 . . . . . 6  |-  ( A  e.  _V  ->  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) )
76biantrud 299 . . . . 5  |-  ( A  e.  _V  ->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. 
<->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  /\  ( ( 1st `  A )  e. 
_V  /\  ( 2nd `  A )  e.  _V ) ) ) )
8 elxp6 5978 . . . . 5  |-  ( A  e.  ( _V  X.  _V )  <->  ( A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) ) )
97, 8syl6rbbr 198 . . . 4  |-  ( A  e.  _V  ->  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
)
109anbi1d 454 . . 3  |-  ( A  e.  _V  ->  (
( 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
) ) ) )
11 elxp6 5978 . . 3  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
1210, 11syl6rbbr 198 . 2  |-  ( A  e.  _V  ->  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A
)  e.  C ) ) ) )
131, 3, 12pm5.21nii 658 1  |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A
)  e.  C ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1296    e. wcel 1445   _Vcvv 2633   <.cop 3469    X. cxp 4465   ` cfv 5049   1stc1st 5947   2ndc2nd 5948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fo 5055  df-fv 5057  df-1st 5949  df-2nd 5950
This theorem is referenced by:  xp2  5981  unielxp  5982  1stconst  6024  2ndconst  6025  f1od2  6038
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