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Theorem cvpss 28528
 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 28525 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
2 simpl 472 . 2 ((𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)) → 𝐴𝐵)
31, 2syl6bi 242 1 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 1977  ∃wrex 2897   ⊊ wpss 3541   class class class wbr 4583   Cℋ cch 27170   ⋖ℋ ccv 27205 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cv 28522 This theorem is referenced by:  cvnsym  28533  cvntr  28535  atcveq0  28591  chcv1  28598  cvati  28609  cvbr4i  28610  cvexchlem  28611  atexch  28624  atcvat2i  28630
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