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Theorem cvnbtwn4 29488
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 29485 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
2 iman 388 . . 3 (((𝐴𝐶𝐶𝐵) → (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)))
3 an4 635 . . . . 5 (((𝐴𝐶𝐶𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵)))
4 ioran 964 . . . . . . 7 (¬ (𝐶 = 𝐴𝐶 = 𝐵) ↔ (¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵))
5 eqcom 2778 . . . . . . . . 9 (𝐶 = 𝐴𝐴 = 𝐶)
65notbii 309 . . . . . . . 8 𝐶 = 𝐴 ↔ ¬ 𝐴 = 𝐶)
76anbi1i 610 . . . . . . 7 ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
84, 7bitri 264 . . . . . 6 (¬ (𝐶 = 𝐴𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
98anbi2i 609 . . . . 5 (((𝐴𝐶𝐶𝐵) ∧ ¬ (𝐶 = 𝐴𝐶 = 𝐵)) ↔ ((𝐴𝐶𝐶𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)))
10 dfpss2 3842 . . . . . 6 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
11 dfpss2 3842 . . . . . 6 (𝐶𝐵 ↔ (𝐶𝐵 ∧ ¬ 𝐶 = 𝐵))
1210, 11anbi12i 612 . . . . 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  wa 382  wo 834  w3a 1071   = wceq 1631  wcel 2145  wss 3723  wpss 3724   class class class wbr 4786   C cch 28126   ccv 28161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-cv 29478
This theorem is referenced by:  cvmdi  29523
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