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Theorem cvpss 30072
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 30069 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
2 simpl 486 . 2 ((𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)) → 𝐴𝐵)
31, 2syl6bi 256 1 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wcel 2112  wrex 3110  wpss 3885   class class class wbr 5033   C cch 28716   ccv 28751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-cv 30066
This theorem is referenced by:  cvnsym  30077  cvntr  30079  atcveq0  30135  chcv1  30142  cvati  30153  cvbr4i  30154  cvexchlem  30155  atexch  30168  atcvat2i  30174
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