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Theorem cvnbtwn3 32170
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 32168 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
2 iman 400 . . 3 (((𝐴𝐶𝐶𝐵) → 𝐴 = 𝐶) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
3 eqcom 2732 . . . 4 (𝐶 = 𝐴𝐴 = 𝐶)
43imbi2i 335 . . 3 (((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴) ↔ ((𝐴𝐶𝐶𝐵) → 𝐴 = 𝐶))
5 dfpss2 4081 . . . . . 6 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
65anbi1i 622 . . . . 5 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶𝐵))
7 an32 644 . . . . 5 (((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
86, 7bitri 274 . . . 4 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
98notbii 319 . . 3 (¬ (𝐴𝐶𝐶𝐵) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
102, 4, 93bitr4ri 303 . 2 (¬ (𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴))
111, 10imbitrdi 250 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wss 3944  wpss 3945   class class class wbr 5149   C cch 30811   ccv 30846
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-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-cv 32161
This theorem is referenced by:  atcveq0  32230  atcvatlem  32267
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