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Theorem cvnsym 28336
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 28331 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
2 cvpss 28331 . . . . 5 ((𝐵C𝐴C ) → (𝐵 𝐴𝐵𝐴))
32ancoms 467 . . . 4 ((𝐴C𝐵C ) → (𝐵 𝐴𝐵𝐴))
4 pssn2lp 3666 . . . . 5 ¬ (𝐵𝐴𝐴𝐵)
54imnani 437 . . . 4 (𝐵𝐴 → ¬ 𝐴𝐵)
63, 5syl6 34 . . 3 ((𝐴C𝐵C ) → (𝐵 𝐴 → ¬ 𝐴𝐵))
76con2d 127 . 2 ((𝐴C𝐵C ) → (𝐴𝐵 → ¬ 𝐵 𝐴))
81, 7syld 45 1 ((𝐴C𝐵C ) → (𝐴 𝐵 → ¬ 𝐵 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wcel 1976  wpss 3537   class class class wbr 4574   C cch 26973   ccv 27008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-br 4575  df-opab 4635  df-cv 28325
This theorem is referenced by:  cvnref  28337
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