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

Theorem cvntr 31532
Description: The covers relation is not transitive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvntr ((𝐴C𝐵C𝐶C ) → ((𝐴 𝐵𝐵 𝐶) → ¬ 𝐴 𝐶))

Proof of Theorem cvntr
StepHypRef Expression
1 cvpss 31525 . . 3 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
213adant3 1132 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵𝐴𝐵))
3 cvpss 31525 . . 3 ((𝐵C𝐶C ) → (𝐵 𝐶𝐵𝐶))
433adant1 1130 . 2 ((𝐴C𝐵C𝐶C ) → (𝐵 𝐶𝐵𝐶))
5 cvnbtwn 31526 . . . 4 ((𝐴C𝐶C𝐵C ) → (𝐴 𝐶 → ¬ (𝐴𝐵𝐵𝐶)))
653com23 1126 . . 3 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐶 → ¬ (𝐴𝐵𝐵𝐶)))
76con2d 134 . 2 ((𝐴C𝐵C𝐶C ) → ((𝐴𝐵𝐵𝐶) → ¬ 𝐴 𝐶))
82, 4, 7syl2and 608 1 ((𝐴C𝐵C𝐶C ) → ((𝐴 𝐵𝐵 𝐶) → ¬ 𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087  wcel 2106  wpss 3948   class class class wbr 5147   C cch 30169   ccv 30204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-cv 31519
This theorem is referenced by:  atcv0eq  31619
  Copyright terms: Public domain W3C validator