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Theorem cvnsym 32048
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 32043 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
2 cvpss 32043 . . . . 5 ((𝐵C𝐴C ) → (𝐵 𝐴𝐵𝐴))
32ancoms 458 . . . 4 ((𝐴C𝐵C ) → (𝐵 𝐴𝐵𝐴))
4 pssn2lp 4096 . . . . 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 2098  wpss 3944   class class class wbr 5141   C cch 30687   ccv 30722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-cv 32037
This theorem is referenced by:  cvnref  32049
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