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Theorem brdifun 6069
Description: Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypothesis
Ref Expression
swoer.1 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
Assertion
Ref Expression
brdifun ((A 𝑋 B 𝑋) → (A𝑅B ↔ ¬ (A < B B < A)))

Proof of Theorem brdifun
StepHypRef Expression
1 opelxpi 4319 . . . 4 ((A 𝑋 B 𝑋) → ⟨A, B (𝑋 × 𝑋))
2 df-br 3756 . . . 4 (A(𝑋 × 𝑋)B ↔ ⟨A, B (𝑋 × 𝑋))
31, 2sylibr 137 . . 3 ((A 𝑋 B 𝑋) → A(𝑋 × 𝑋)B)
4 swoer.1 . . . . . 6 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
54breqi 3761 . . . . 5 (A𝑅BA((𝑋 × 𝑋) ∖ ( < < ))B)
6 brdif 3803 . . . . 5 (A((𝑋 × 𝑋) ∖ ( < < ))B ↔ (A(𝑋 × 𝑋)B ¬ A( < < )B))
75, 6bitri 173 . . . 4 (A𝑅B ↔ (A(𝑋 × 𝑋)B ¬ A( < < )B))
87baib 827 . . 3 (A(𝑋 × 𝑋)B → (A𝑅B ↔ ¬ A( < < )B))
93, 8syl 14 . 2 ((A 𝑋 B 𝑋) → (A𝑅B ↔ ¬ A( < < )B))
10 brun 3801 . . . 4 (A( < < )B ↔ (A < B A < B))
11 brcnvg 4459 . . . . 5 ((A 𝑋 B 𝑋) → (A < BB < A))
1211orbi2d 703 . . . 4 ((A 𝑋 B 𝑋) → ((A < B A < B) ↔ (A < B B < A)))
1310, 12syl5bb 181 . . 3 ((A 𝑋 B 𝑋) → (A( < < )B ↔ (A < B B < A)))
1413notbid 591 . 2 ((A 𝑋 B 𝑋) → (¬ A( < < )B ↔ ¬ (A < B B < A)))
159, 14bitrd 177 1 ((A 𝑋 B 𝑋) → (A𝑅B ↔ ¬ (A < B B < A)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   = wceq 1242   wcel 1390  cdif 2908  cun 2909  cop 3370   class class class wbr 3755   × cxp 4286  ccnv 4287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296
This theorem is referenced by:  swoer  6070  swoord1  6071  swoord2  6072
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