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Theorem cvnbtwn3 30659
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 30657 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
2 iman 402 . . 3 (((𝐴𝐶𝐶𝐵) → 𝐴 = 𝐶) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
3 eqcom 2747 . . . 4 (𝐶 = 𝐴𝐴 = 𝐶)
43imbi2i 336 . . 3 (((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴) ↔ ((𝐴𝐶𝐶𝐵) → 𝐴 = 𝐶))
5 dfpss2 4025 . . . . . 6 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
65anbi1i 624 . . . . 5 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶𝐵))
7 an32 643 . . . . 5 (((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
86, 7bitri 274 . . . 4 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
98notbii 320 . . 3 (¬ (𝐴𝐶𝐶𝐵) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
102, 4, 93bitr4ri 304 . 2 (¬ (𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴))
111, 10syl6ib 250 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1086   = wceq 1542  wcel 2110  wss 3892  wpss 3893   class class class wbr 5079   C cch 29300   ccv 29335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-cv 30650
This theorem is referenced by:  atcveq0  30719  atcvatlem  30756
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