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Theorem cvnbtwn2 32049
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 32048 . 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 4080 . . . . . 6 (𝐶𝐵 ↔ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵))
54anbi2i 622 . . . . 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 250 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wss 3943  wpss 3944   class class class wbr 5141   C cch 30691   ccv 30726
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-cv 32041
This theorem is referenced by:  cvati  32128  cvexchlem  32130  atexch  32143
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