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Theorem weinxp 5076
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 5074 . . 3  |-  ( R  Fr  A  <->  ( R  i^i  ( A  X.  A
) )  Fr  A
)
2 soinxp 5073 . . 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 4849 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  R  Or  A ) )
5 df-we 4849 . 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 3470    Or wor 4808    Fr wfr 4844    We wwe 4846    X. cxp 5006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-xp 5014
This theorem is referenced by:  wemapwe  8156  wemapweOLD  8157  infxpenlem  8408  dfac8b  8429  ac10ct  8432  canthwelem  9045  ltbwe  18263  vitali  22147  fin2so  30202  dnwech  31156  aomclem5  31166
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