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Theorem cvpss 32313
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 32310 . 2 ((𝐴C𝐵C ) → (𝐴 𝐵 ↔ (𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵))))
2 simpl 482 . 2 ((𝐴𝐵 ∧ ¬ ∃𝑥C (𝐴𝑥𝑥𝐵)) → 𝐴𝐵)
31, 2biimtrdi 253 1 ((𝐴C𝐵C ) → (𝐴 𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wcel 2105  wrex 3067  wpss 3963   class class class wbr 5147   C cch 30957   ccv 30992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-cv 32307
This theorem is referenced by:  cvnsym  32318  cvntr  32320  atcveq0  32376  chcv1  32383  cvati  32394  cvbr4i  32395  cvexchlem  32396  atexch  32409  atcvat2i  32415
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