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Theorem elelpwi 4204
Description: If 𝐴 belongs to a part of 𝐶 then 𝐴 belongs to 𝐶. (Contributed by FL, 3-Aug-2009.)
Assertion
Ref Expression
elelpwi ((𝐴𝐵𝐵 ∈ 𝒫 𝐶) → 𝐴𝐶)

Proof of Theorem elelpwi
StepHypRef Expression
1 elpwi 4201 . . 3 (𝐵 ∈ 𝒫 𝐶𝐵𝐶)
21sseld 3635 . 2 (𝐵 ∈ 𝒫 𝐶 → (𝐴𝐵𝐴𝐶))
32impcom 445 1 ((𝐴𝐵𝐵 ∈ 𝒫 𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  𝒫 cpw 4191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193
This theorem is referenced by:  unipw  4948  axdc2lem  9308  axdc3lem4  9313  homarel  16733  txdis  21483  uhgredgrnv  26070  fpwrelmap  29636  insiga  30328  measinblem  30411  ddemeas  30427  imambfm  30452  totprobd  30616  dstrvprob  30661  ballotlem2  30678  scmsuppss  42478  lincvalsc0  42535  linc0scn0  42537  lincdifsn  42538  linc1  42539  lincsum  42543  lincscm  42544  lcoss  42550  lincext3  42570  islindeps2  42597
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