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Theorem cvnbtwn3 28531
Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnbtwn3 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴)))

Proof of Theorem cvnbtwn3
StepHypRef Expression
1 cvnbtwn 28529 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
2 iman 439 . . 3 (((𝐴𝐶𝐶𝐵) → 𝐴 = 𝐶) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
3 eqcom 2617 . . . 4 (𝐶 = 𝐴𝐴 = 𝐶)
43imbi2i 325 . . 3 (((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴) ↔ ((𝐴𝐶𝐶𝐵) → 𝐴 = 𝐶))
5 dfpss2 3654 . . . . . 6 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
65anbi1i 727 . . . . 5 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶𝐵))
7 an32 835 . . . . 5 (((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
86, 7bitri 263 . . . 4 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
98notbii 309 . . 3 (¬ (𝐴𝐶𝐶𝐵) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
102, 4, 93bitr4ri 292 . 2 (¬ (𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴))
111, 10syl6ib 240 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wss 3540  wpss 3541   class class class wbr 4583   C cch 27170   ccv 27205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cv 28522
This theorem is referenced by:  atcveq0  28591  atcvatlem  28628
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