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Theorem elelpwi 4119
 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 4117 . . 3 (𝐵 ∈ 𝒫 𝐶𝐵𝐶)
21sseld 3567 . 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 1977  𝒫 cpw 4108 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110 This theorem is referenced by:  unipw  4845  axdc2lem  9153  axdc3lem4  9158  homarel  16509  txdis  21245  uhgredgrnv  25804  fpwrelmap  28896  insiga  29527  measinblem  29610  ddemeas  29626  imambfm  29651  totprobd  29815  dstrvprob  29860  ballotlem2  29877  scmsuppss  41947  lincvalsc0  42004  linc0scn0  42006  lincdifsn  42007  linc1  42008  lincsum  42012  lincscm  42013  lcoss  42019  lincext3  42039  islindeps2  42066
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