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Theorem cvnsym 29037
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 29032 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
2 cvpss 29032 . . . . 5 ((𝐵C𝐴C ) → (𝐵 𝐴𝐵𝐴))
32ancoms 469 . . . 4 ((𝐴C𝐵C ) → (𝐵 𝐴𝐵𝐴))
4 pssn2lp 3692 . . . . 5 ¬ (𝐵𝐴𝐴𝐵)
54imnani 439 . . . 4 (𝐵𝐴 → ¬ 𝐴𝐵)
63, 5syl6 35 . . 3 ((𝐴C𝐵C ) → (𝐵 𝐴 → ¬ 𝐴𝐵))
76con2d 129 . 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 384  wcel 1987  wpss 3561   class class class wbr 4623   C cch 27674   ccv 27709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-cv 29026
This theorem is referenced by:  cvnref  29038
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