HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  cvnbtwn3 Structured version   Visualization version   GIF version

Theorem cvnbtwn3 28987
Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnbtwn3 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴)))

Proof of Theorem cvnbtwn3
StepHypRef Expression
1 cvnbtwn 28985 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ¬ (𝐴𝐶𝐶𝐵)))
2 iman 440 . . 3 (((𝐴𝐶𝐶𝐵) → 𝐴 = 𝐶) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
3 eqcom 2633 . . . 4 (𝐶 = 𝐴𝐴 = 𝐶)
43imbi2i 326 . . 3 (((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴) ↔ ((𝐴𝐶𝐶𝐵) → 𝐴 = 𝐶))
5 dfpss2 3675 . . . . . 6 (𝐴𝐶 ↔ (𝐴𝐶 ∧ ¬ 𝐴 = 𝐶))
65anbi1i 730 . . . . 5 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶𝐵))
7 an32 838 . . . . 5 (((𝐴𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
86, 7bitri 264 . . . 4 ((𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
98notbii 310 . . 3 (¬ (𝐴𝐶𝐶𝐵) ↔ ¬ ((𝐴𝐶𝐶𝐵) ∧ ¬ 𝐴 = 𝐶))
102, 4, 93bitr4ri 293 . 2 (¬ (𝐴𝐶𝐶𝐵) ↔ ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴))
111, 10syl6ib 241 1 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵 → ((𝐴𝐶𝐶𝐵) → 𝐶 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  wss 3560  wpss 3561   class class class wbr 4618   C cch 27626   ccv 27661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-cv 28978
This theorem is referenced by:  atcveq0  29047  atcvatlem  29084
  Copyright terms: Public domain W3C validator