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Theorem cvnbtwn4 28338
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 28335 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
2 iman 438 . . 3 (((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)))
3 an4 860 . . . . 5 (((𝐴𝐶𝐶𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵)))
4 ioran 509 . . . . . . 7 (¬ (𝐶 = 𝐴𝐶 = 𝐵) ↔ (¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵))
5 eqcom 2616 . . . . . . . . 9 (𝐶 = 𝐴𝐴 = 𝐶)
65notbii 308 . . . . . . . 8 𝐶 = 𝐴 ↔ ¬ 𝐴 = 𝐶)
76anbi1i 726 . . . . . . 7 ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
84, 7bitri 262 . . . . . 6 (¬ (𝐶 = 𝐴𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
98anbi2i 725 . . . . 5 (((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ((𝐴𝐶𝐶𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)))
10 dfpss2 3653 . . . . . 6 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
11 dfpss2 3653 . . . . . 6 (𝐶𝐵 ↔ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵))
1210, 11anbi12i 728 . . . . 5 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵)))
133, 9, 123bitr4i 290 . . . 4 (((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)) ↔ (𝐴𝐶𝐶𝐵))
1413notbii 308 . . 3 (¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ¬ (𝐴𝐶𝐶𝐵))
152, 14bitr2i 263 . 2 (¬ (𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵)))
161, 15syl6ib 239 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  wss 3539  wpss 3540   class class class wbr 4577   C cch 26976   ccv 27011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-cv 28328
This theorem is referenced by:  cvmdi  28373
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