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Theorem cvpss 32094
Description: The covers relation implies proper subset. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)
Assertion
Ref Expression
cvpss ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))

Proof of Theorem cvpss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cvbr 32091 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
2 simpl 482 . 2 ((𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)) → 𝐴𝐵)
31, 2biimtrdi 252 1 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2099  wrex 3067  wpss 3948   class class class wbr 5148   C cch 30738   ccv 30773
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 32088
This theorem is referenced by:  cvnsym  32099  cvntr  32101  atcveq0  32157  chcv1  32164  cvati  32175  cvbr4i  32176  cvexchlem  32177  atexch  32190  atcvat2i  32196
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