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Theorem elxp7 5825
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4836. (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 2583 . 2  |-  ( A  e.  ( B  X.  C )  ->  A  e.  _V )
2 elex 2583 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  A  e. 
_V )
32adantr 265 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  A  e.  _V )
4 1stexg 5822 . . . . . . 7  |-  ( A  e.  _V  ->  ( 1st `  A )  e. 
_V )
5 2ndexg 5823 . . . . . . 7  |-  ( A  e.  _V  ->  ( 2nd `  A )  e. 
_V )
64, 5jca 294 . . . . . 6  |-  ( A  e.  _V  ->  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) )
76biantrud 292 . . . . 5  |-  ( A  e.  _V  ->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. 
<->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  /\  ( ( 1st `  A )  e. 
_V  /\  ( 2nd `  A )  e.  _V ) ) ) )
8 elxp6 5824 . . . . 5  |-  ( A  e.  ( _V  X.  _V )  <->  ( A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) ) )
97, 8syl6rbbr 192 . . . 4  |-  ( A  e.  _V  ->  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
)
109anbi1d 446 . . 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 5824 . . 3  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
1210, 11syl6rbbr 192 . 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 630 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 101    <-> wb 102    = wceq 1259    e. wcel 1409   _Vcvv 2574   <.cop 3406    X. cxp 4371   ` cfv 4930   1stc1st 5793   2ndc2nd 5794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-1st 5795  df-2nd 5796
This theorem is referenced by:  xp2  5827  unielxp  5828  1stconst  5870  2ndconst  5871  f1od2  5884
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