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Theorem cvntr 28328
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 28321 . . 3 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
213adant3 1073 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵𝐴𝐵))
3 cvpss 28321 . . 3 ((𝐵C𝐶C ) → (𝐵 𝐶𝐵𝐶))
433adant1 1071 . 2 ((𝐴C𝐵C𝐶C ) → (𝐵 𝐶𝐵𝐶))
5 cvnbtwn 28322 . . . 4 ((𝐴C𝐶C𝐵C ) → (𝐴 𝐶 → ¬ (𝐴𝐵𝐵𝐶)))
653com23 1262 . . 3 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐶 → ¬ (𝐴𝐵𝐵𝐶)))
76con2d 127 . 2 ((𝐴C𝐵C𝐶C ) → ((𝐴𝐵𝐵𝐶) → ¬ 𝐴 𝐶))
82, 4, 7syl2and 498 1 ((𝐴C𝐵C𝐶C ) → ((𝐴 𝐵𝐵 𝐶) → ¬ 𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1030  wcel 1976  wpss 3540   class class class wbr 4577   C cch 26963   ccv 26998
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 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
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 28315
This theorem is referenced by:  atcv0eq  28415
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