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Theorem cvnbtwn4 30058
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 30055 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
2 iman 404 . . 3 (((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)))
3 an4 654 . . . . 5 (((𝐴𝐶𝐶𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵)))
4 ioran 979 . . . . . . 7 (¬ (𝐶 = 𝐴𝐶 = 𝐵) ↔ (¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵))
5 eqcom 2826 . . . . . . . . 9 (𝐶 = 𝐴𝐴 = 𝐶)
65notbii 322 . . . . . . . 8 𝐶 = 𝐴 ↔ ¬ 𝐴 = 𝐶)
76anbi1i 625 . . . . . . 7 ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
84, 7bitri 277 . . . . . 6 (¬ (𝐶 = 𝐴𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
98anbi2i 624 . . . . 5 (((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ((𝐴𝐶𝐶𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)))
10 dfpss2 4060 . . . . . 6 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
11 dfpss2 4060 . . . . . 6 (𝐶𝐵 ↔ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵))
1210, 11anbi12i 628 . . . . 5 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵)))
133, 9, 123bitr4i 305 . . . 4 (((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)) ↔ (𝐴𝐶𝐶𝐵))
1413notbii 322 . . 3 (¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ¬ (𝐴𝐶𝐶𝐵))
152, 14bitr2i 278 . 2 (¬ (𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵)))
161, 15syl6ib 253 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wo 843  w3a 1081   = wceq 1530  wcel 2107  wss 3934  wpss 3935   class class class wbr 5057   C cch 28698   ccv 28733
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-cv 30048
This theorem is referenced by:  cvmdi  30093
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