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Theorem weinxp 5066
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 5064 . . 3  |-  ( R  Fr  A  <->  ( R  i^i  ( A  X.  A
) )  Fr  A
)
2 soinxp 5063 . . 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 4840 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  R  Or  A ) )
5 df-we 4840 . 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 3475    Or wor 4799    Fr wfr 4835    We wwe 4837    X. cxp 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-xp 5005
This theorem is referenced by:  wemapwe  8135  wemapweOLD  8136  infxpenlem  8387  dfac8b  8408  ac10ct  8411  canthwelem  9024  ltbwe  17905  vitali  21754  fin2so  29614  dnwech  30598  aomclem5  30608
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