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Theorem cvntr 30942
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 30935 . . 3 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
213adant3 1131 . 2 ((𝐴C𝐵C𝐶C ) → (𝐴 𝐵𝐴𝐵))
3 cvpss 30935 . . 3 ((𝐵C𝐶C ) → (𝐵 𝐶𝐵𝐶))
433adant1 1129 . 2 ((𝐴C𝐵C𝐶C ) → (𝐵 𝐶𝐵𝐶))
5 cvnbtwn 30936 . . . 4 ((𝐴C𝐶C𝐵C ) → (𝐴 𝐶 → ¬ (𝐴𝐵𝐵𝐶)))
653com23 1125 . . 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 1086  wcel 2105  wpss 3899   class class class wbr 5092   C cch 29579   ccv 29614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-cv 30929
This theorem is referenced by:  atcv0eq  31029
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