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Theorem cvnbtwn4 29276
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 29273 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
2 iman 439 . . 3 (((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)))
3 an4 882 . . . . 5 (((𝐴𝐶𝐶𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵)))
4 ioran 510 . . . . . . 7 (¬ (𝐶 = 𝐴𝐶 = 𝐵) ↔ (¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵))
5 eqcom 2658 . . . . . . . . 9 (𝐶 = 𝐴𝐴 = 𝐶)
65notbii 309 . . . . . . . 8 𝐶 = 𝐴 ↔ ¬ 𝐴 = 𝐶)
76anbi1i 731 . . . . . . 7 ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
84, 7bitri 264 . . . . . 6 (¬ (𝐶 = 𝐴𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
98anbi2i 730 . . . . 5 (((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ((𝐴𝐶𝐶𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)))
10 dfpss2 3725 . . . . . 6 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
11 dfpss2 3725 . . . . . 6 (𝐶𝐵 ↔ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵))
1210, 11anbi12i 733 . . . . 5 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵)))
133, 9, 123bitr4i 292 . . . 4 (((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)) ↔ (𝐴𝐶𝐶𝐵))
1413notbii 309 . . 3 (¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ¬ (𝐴𝐶𝐶𝐵))
152, 14bitr2i 265 . 2 (¬ (𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵)))
161, 15syl6ib 241 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wss 3607  wpss 3608   class class class wbr 4685   C cch 27914   ccv 27949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-cv 29266
This theorem is referenced by:  cvmdi  29311
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