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Theorem elelpwi 4142
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 4140 . . 3 (𝐵 ∈ 𝒫 𝐶𝐵𝐶)
21sseld 3582 . 2 (𝐵 ∈ 𝒫 𝐶 → (𝐴𝐵𝐴𝐶))
32impcom 446 1 ((𝐴𝐵𝐵 ∈ 𝒫 𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  𝒫 cpw 4130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-in 3562  df-ss 3569  df-pw 4132
This theorem is referenced by:  unipw  4879  axdc2lem  9214  axdc3lem4  9219  homarel  16607  txdis  21345  uhgredgrnv  25920  fpwrelmap  29348  insiga  29978  measinblem  30061  ddemeas  30077  imambfm  30102  totprobd  30266  dstrvprob  30311  ballotlem2  30328  scmsuppss  41438  lincvalsc0  41495  linc0scn0  41497  lincdifsn  41498  linc1  41499  lincsum  41503  lincscm  41504  lcoss  41510  lincext3  41530  islindeps2  41557
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