HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  cvnbtwn2 Structured version   Visualization version   GIF version

Theorem cvnbtwn2 32316
Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnbtwn2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐵)))

Proof of Theorem cvnbtwn2
StepHypRef Expression
1 cvnbtwn 32315 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
2 iman 401 . . 3 (((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐵) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐶 = 𝐵))
3 anass 468 . . . . 5 (((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴𝐶 ∧ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵)))
4 dfpss2 4098 . . . . . 6 (𝐶𝐵 ↔ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵))
54anbi2i 623 . . . . 5 ((𝐴𝐶𝐶𝐵) ↔ (𝐴𝐶 ∧ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵)))
63, 5bitr4i 278 . . . 4 (((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴𝐶𝐶𝐵))
76notbii 320 . . 3 (¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ ¬ (𝐴𝐶𝐶𝐵))
82, 7bitr2i 276 . 2 (¬ (𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐵))
91, 8imbitrdi 251 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wss 3963  wpss 3964   class class class wbr 5148   C cch 30958   ccv 30993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cv 32308
This theorem is referenced by:  cvati  32395  cvexchlem  32397  atexch  32410
  Copyright terms: Public domain W3C validator