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Theorem cvnsym 30071
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 30066 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
2 cvpss 30066 . . . . 5 ((𝐵C𝐴C ) → (𝐵 𝐴𝐵𝐴))
32ancoms 462 . . . 4 ((𝐴C𝐵C ) → (𝐵 𝐴𝐵𝐴))
4 pssn2lp 4053 . . . . 5 ¬ (𝐵𝐴𝐴𝐵)
54imnani 404 . . . 4 (𝐵𝐴 → ¬ 𝐴𝐵)
63, 5syl6 35 . . 3 ((𝐴C𝐵C ) → (𝐵 𝐴 → ¬ 𝐴𝐵))
76con2d 136 . 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 399  wcel 2114  wpss 3909   class class class wbr 5042   C cch 28710   ccv 28745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-cv 30060
This theorem is referenced by:  cvnref  30072
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