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Theorem cvnsym 32113
Description: The covers relation is not symmetric. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvnsym ((𝐴C𝐵C ) → (𝐴 𝐵 → ¬ 𝐵 𝐴))

Proof of Theorem cvnsym
StepHypRef Expression
1 cvpss 32108 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
2 cvpss 32108 . . . . 5 ((𝐵C𝐴C ) → (𝐵 𝐴𝐵𝐴))
32ancoms 458 . . . 4 ((𝐴C𝐵C ) → (𝐵 𝐴𝐵𝐴))
4 pssn2lp 4099 . . . . 5 ¬ (𝐵𝐴𝐴𝐵)
54imnani 400 . . . 4 (𝐵𝐴 → ¬ 𝐴𝐵)
63, 5syl6 35 . . 3 ((𝐴C𝐵C ) → (𝐵 𝐴 → ¬ 𝐴𝐵))
76con2d 134 . 2 ((𝐴C𝐵C ) → (𝐴𝐵 → ¬ 𝐵 𝐴))
81, 7syld 47 1 ((𝐴C𝐵C ) → (𝐴 𝐵 → ¬ 𝐵 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2099  wpss 3948   class class class wbr 5148   C cch 30752   ccv 30787
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 5299  ax-nul 5306  ax-pr 5429
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 2938  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-cv 32102
This theorem is referenced by:  cvnref  32114
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