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Theorem weinxp 4904
Description: Intersection of well-ordering with Cartesian product of its field. (Contributed by NM, 9-Mar-1997.) (Revised by Mario Carneiro, 10-Jul-2014.)
Assertion
Ref Expression
weinxp  |-  ( R  We  A  <->  ( R  i^i  ( A  X.  A
) )  We  A
)

Proof of Theorem weinxp
StepHypRef Expression
1 frinxp 4902 . . 3  |-  ( R  Fr  A  <->  ( R  i^i  ( A  X.  A
) )  Fr  A
)
2 soinxp 4901 . . 3  |-  ( R  Or  A  <->  ( R  i^i  ( A  X.  A
) )  Or  A
)
31, 2anbi12i 697 . 2  |-  ( ( R  Fr  A  /\  R  Or  A )  <->  ( ( R  i^i  ( A  X.  A ) )  Fr  A  /\  ( R  i^i  ( A  X.  A ) )  Or  A ) )
4 df-we 4679 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  R  Or  A ) )
5 df-we 4679 . 2  |-  ( ( R  i^i  ( A  X.  A ) )  We  A  <->  ( ( R  i^i  ( A  X.  A ) )  Fr  A  /\  ( R  i^i  ( A  X.  A ) )  Or  A ) )
63, 4, 53bitr4i 277 1  |-  ( R  We  A  <->  ( R  i^i  ( A  X.  A
) )  We  A
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    i^i cin 3325    Or wor 4638    Fr wfr 4674    We wwe 4676    X. cxp 4836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-br 4291  df-opab 4349  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-xp 4844
This theorem is referenced by:  wemapwe  7926  wemapweOLD  7927  infxpenlem  8178  dfac8b  8199  ac10ct  8202  canthwelem  8815  ltbwe  17552  vitali  21091  fin2so  28413  dnwech  29398  aomclem5  29408
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