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Theorem cvntr 32095
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 32088 . . 3 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
213adant3 1130 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵𝐴𝐵))
3 cvpss 32088 . . 3 ((𝐵C𝐶C ) → (𝐵 𝐶𝐵𝐶))
433adant1 1128 . 2 ((𝐴C𝐵C𝐶C ) → (𝐵 𝐶𝐵𝐶))
5 cvnbtwn 32089 . . . 4 ((𝐴C𝐶C𝐵C ) → (𝐴 𝐶 → ¬ (𝐴𝐵𝐵𝐶)))
653com23 1124 . . 3 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐶 → ¬ (𝐴𝐵𝐵𝐶)))
76con2d 134 . 2 ((𝐴C𝐵C𝐶C ) → ((𝐴𝐵𝐵𝐶) → ¬ 𝐴 𝐶))
82, 4, 7syl2and 607 1 ((𝐴C𝐵C𝐶C ) → ((𝐴 𝐵𝐵 𝐶) → ¬ 𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1085  wcel 2099  wpss 3946   class class class wbr 5142   C cch 30732   ccv 30767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-cv 32082
This theorem is referenced by:  atcv0eq  32182
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