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Theorem cvpss 32032
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 32029 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
2 simpl 482 . 2 ((𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)) → 𝐴𝐵)
31, 2syl6bi 253 1 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2098  wrex 3062  wpss 3942   class class class wbr 5139   C cch 30676   ccv 30711
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 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
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 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-cv 32026
This theorem is referenced by:  cvnsym  32037  cvntr  32039  atcveq0  32095  chcv1  32102  cvati  32113  cvbr4i  32114  cvexchlem  32115  atexch  32128  atcvat2i  32134
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