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Theorem cvnbtwn4 31273
Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnbtwn4 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵))))

Proof of Theorem cvnbtwn4
StepHypRef Expression
1 cvnbtwn 31270 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
2 iman 403 . . 3 (((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)))
3 an4 655 . . . . 5 (((𝐴𝐶𝐶𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵)))
4 ioran 983 . . . . . . 7 (¬ (𝐶 = 𝐴𝐶 = 𝐵) ↔ (¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵))
5 eqcom 2740 . . . . . . . . 9 (𝐶 = 𝐴𝐴 = 𝐶)
65notbii 320 . . . . . . . 8 𝐶 = 𝐴 ↔ ¬ 𝐴 = 𝐶)
76anbi1i 625 . . . . . . 7 ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
84, 7bitri 275 . . . . . 6 (¬ (𝐶 = 𝐴𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
98anbi2i 624 . . . . 5 (((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ((𝐴𝐶𝐶𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)))
10 dfpss2 4046 . . . . . 6 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
11 dfpss2 4046 . . . . . 6 (𝐶𝐵 ↔ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵))
1210, 11anbi12i 628 . . . . 5 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵)))
133, 9, 123bitr4i 303 . . . 4 (((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)) ↔ (𝐴𝐶𝐶𝐵))
1413notbii 320 . . 3 (¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ¬ (𝐴𝐶𝐶𝐵))
152, 14bitr2i 276 . 2 (¬ (𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵)))
161, 15syl6ib 251 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wss 3911  wpss 3912   class class class wbr 5106   C cch 29913   ccv 29948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-cv 31263
This theorem is referenced by:  cvmdi  31308
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