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Theorem weinxp 5629
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 ∩ (𝐴 × 𝐴)) We 𝐴)

Proof of Theorem weinxp
StepHypRef Expression
1 frinxp 5627 . . 3 (𝑅 Fr 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴)
2 soinxp 5626 . . 3 (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)
31, 2anbi12i 626 . 2 ((𝑅 Fr 𝐴𝑅 Or 𝐴) ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴 ∧ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴))
4 df-we 5509 . 2 (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴𝑅 Or 𝐴))
5 df-we 5509 . 2 ((𝑅 ∩ (𝐴 × 𝐴)) We 𝐴 ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Fr 𝐴 ∧ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴))
63, 4, 53bitr4i 304 1 (𝑅 We 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) We 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  cin 3932   Or wor 5466   Fr wfr 5504   We wwe 5506   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554
This theorem is referenced by:  wemapwe  9148  infxpenlem  9427  dfac8b  9445  ac10ct  9448  canthwelem  10060  ltbwe  20181  vitali  24141  fin2so  34760  dnwech  39526  aomclem5  39536
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