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Theorem brdifun 6040
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 4299 . . . 4 ((A 𝑋 B 𝑋) → ⟨A, B (𝑋 × 𝑋))
2 df-br 3735 . . . 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 3740 . . . . 5 (A𝑅BA((𝑋 × 𝑋) ∖ ( < < ))B)
6 brdif 3782 . . . . 5 (A((𝑋 × 𝑋) ∖ ( < < ))B ↔ (A(𝑋 × 𝑋)B ¬ A( < < )B))
75, 6bitri 173 . . . 4 (A𝑅B ↔ (A(𝑋 × 𝑋)B ¬ A( < < )B))
87baib 816 . . 3 (A(𝑋 × 𝑋)B → (A𝑅B ↔ ¬ A( < < )B))
93, 8syl 14 . 2 ((A 𝑋 B 𝑋) → (A𝑅B ↔ ¬ A( < < )B))
10 brun 3780 . . . 4 (A( < < )B ↔ (A < B A < B))
11 brcnvg 4439 . . . . 5 ((A 𝑋 B 𝑋) → (A < BB < A))
1211orbi2d 691 . . . 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 579 . 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 616   = wceq 1226   wcel 1370  cdif 2887  cun 2888  cop 3349   class class class wbr 3734   × cxp 4266  ccnv 4267
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-cnv 4276
This theorem is referenced by:  swoer  6041  swoord1  6042  swoord2  6043
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