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Theorem elxp7 6185
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5128. (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 2760 . 2  |-  ( A  e.  ( B  X.  C )  ->  A  e.  _V )
2 elex 2760 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  A  e. 
_V )
32adantr 276 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  A  e.  _V )
4 elxp6 6184 . . 3  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
5 elxp6 6184 . . . . 5  |-  ( A  e.  ( _V  X.  _V )  <->  ( A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) ) )
6 1stexg 6182 . . . . . . 7  |-  ( A  e.  _V  ->  ( 1st `  A )  e. 
_V )
7 2ndexg 6183 . . . . . . 7  |-  ( A  e.  _V  ->  ( 2nd `  A )  e. 
_V )
86, 7jca 306 . . . . . 6  |-  ( A  e.  _V  ->  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) )
98biantrud 304 . . . . 5  |-  ( A  e.  _V  ->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. 
<->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  /\  ( ( 1st `  A )  e. 
_V  /\  ( 2nd `  A )  e.  _V ) ) ) )
105, 9bitr4id 199 . . . 4  |-  ( A  e.  _V  ->  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
)
1110anbi1d 465 . . 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
) ) ) )
124, 11bitr4id 199 . 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 705 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 104    <-> wb 105    = wceq 1363    e. wcel 2158   _Vcvv 2749   <.cop 3607    X. cxp 4636   ` cfv 5228   1stc1st 6153   2ndc2nd 6154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fo 5234  df-fv 5236  df-1st 6155  df-2nd 6156
This theorem is referenced by:  xp2  6188  unielxp  6189  1stconst  6236  2ndconst  6237  f1od2  6250
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