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Theorem cvnbtwn3 29993
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 29991 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
2 iman 402 . . 3 (((𝐴𝐶𝐶𝐵) → 𝐴 = 𝐶) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
3 eqcom 2828 . . . 4 (𝐶 = 𝐴𝐴 = 𝐶)
43imbi2i 337 . . 3 (((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴) ↔ ((𝐴𝐶𝐶𝐵) → 𝐴 = 𝐶))
5 dfpss2 4061 . . . . . 6 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
65anbi1i 623 . . . . 5 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶𝐵))
7 an32 642 . . . . 5 (((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
86, 7bitri 276 . . . 4 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
98notbii 321 . . 3 (¬ (𝐴𝐶𝐶𝐵) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
102, 4, 93bitr4ri 305 . 2 (¬ (𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴))
111, 10syl6ib 252 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wss 3935  wpss 3936   class class class wbr 5058   C cch 28634   ccv 28669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5059  df-opab 5121  df-cv 29984
This theorem is referenced by:  atcveq0  30053  atcvatlem  30090
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