MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brdifun Structured version   Visualization version   GIF version

Theorem brdifun 8754
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 5715 . . . 4 ((𝐴𝑋𝐵𝑋) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
2 df-br 5150 . . . 4 (𝐴(𝑋 × 𝑋)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑋))
31, 2sylibr 233 . . 3 ((𝐴𝑋𝐵𝑋) → 𝐴(𝑋 × 𝑋)𝐵)
4 swoer.1 . . . . . 6 𝑅 = ((𝑋 × 𝑋) ∖ ( < < ))
54breqi 5155 . . . . 5 (𝐴𝑅𝐵𝐴((𝑋 × 𝑋) ∖ ( < < ))𝐵)
6 brdif 5202 . . . . 5 (𝐴((𝑋 × 𝑋) ∖ ( < < ))𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < < )𝐵))
75, 6bitri 274 . . . 4 (𝐴𝑅𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < < )𝐵))
87baib 534 . . 3 (𝐴(𝑋 × 𝑋)𝐵 → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < < )𝐵))
93, 8syl 17 . 2 ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < < )𝐵))
10 brun 5200 . . . 4 (𝐴( < < )𝐵 ↔ (𝐴 < 𝐵𝐴 < 𝐵))
11 brcnvg 5882 . . . . 5 ((𝐴𝑋𝐵𝑋) → (𝐴 < 𝐵𝐵 < 𝐴))
1211orbi2d 913 . . . 4 ((𝐴𝑋𝐵𝑋) → ((𝐴 < 𝐵𝐴 < 𝐵) ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
1310, 12bitrid 282 . . 3 ((𝐴𝑋𝐵𝑋) → (𝐴( < < )𝐵 ↔ (𝐴 < 𝐵𝐵 < 𝐴)))
1413notbid 317 . 2 ((𝐴𝑋𝐵𝑋) → (¬ 𝐴( < < )𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
159, 14bitrd 278 1 ((𝐴𝑋𝐵𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵𝐵 < 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wcel 2098  cdif 3941  cun 3942  cop 4636   class class class wbr 5149   × cxp 5676  ccnv 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686
This theorem is referenced by:  swoer  8755  swoord1  8756  swoord2  8757  swoso  8758
  Copyright terms: Public domain W3C validator