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Theorem brdifun 6463
Description: Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypothesis
Ref Expression
swoer.1 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
Assertion
Ref Expression
brdifun ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))

Proof of Theorem brdifun
StepHypRef Expression
1 opelxpi 4578 . . . 4 ((𝐴𝑋𝐵𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
2 df-br 3937 . . . 4 (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
31, 2sylibr 133 . . 3 ((𝐴𝑋𝐵𝑋) → 𝐴(𝑋 × 𝑋)𝐵)
4 swoer.1 . . . . . 6 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
54breqi 3942 . . . . 5 (𝐴𝑅𝐵𝐴((𝑋 × 𝑋) ∖ ( < < ))𝐵)
6 brdif 3988 . . . . 5 (𝐴((𝑋 × 𝑋) ∖ ( < < ))𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < < )𝐵))
75, 6bitri 183 . . . 4 (𝐴𝑅𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < < )𝐵))
87baib 905 . . 3 (𝐴(𝑋 × 𝑋)𝐵 → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < < )𝐵))
93, 8syl 14 . 2 ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < < )𝐵))
10 brun 3986 . . . 4 (𝐴( < < )𝐵 ↔ (𝐴 < 𝐵𝐴 < 𝐵))
11 brcnvg 4727 . . . . 5 ((𝐴𝑋𝐵𝑋) → (𝐴 < 𝐵𝐵 < 𝐴))
1211orbi2d 780 . . . 4 ((𝐴𝑋𝐵𝑋) → ((𝐴 < 𝐵𝐴 < 𝐵) ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
1310, 12syl5bb 191 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴( < < )𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
1413notbid 657 . 2 ((𝐴𝑋𝐵𝑋) → (¬ 𝐴( < < )𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
159, 14bitrd 187 1 ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698   = wceq 1332  wcel 1481  cdif 3072  cun 3073  cop 3534   class class class wbr 3936   × cxp 4544  ccnv 4545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-xp 4552  df-cnv 4554
This theorem is referenced by:  swoer  6464  swoord1  6465  swoord2  6466
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