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Theorem elelpwi 4611
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 4608 . . 3 (𝐵 ∈ 𝒫 𝐶𝐵𝐶)
21sseld 3980 . 2 (𝐵 ∈ 𝒫 𝐶 → (𝐴𝐵𝐴𝐶))
32impcom 408 1 ((𝐴𝐵𝐵 ∈ 𝒫 𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  𝒫 cpw 4601
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-ss 3964  df-pw 4603
This theorem is referenced by:  unipw  5449  axdc2lem  10439  axdc3lem4  10444  homarel  17982  txdis  23127  uhgredgrnv  28379  fpwrelmap  31945  insiga  33123  measinblem  33206  ddemeas  33222  imambfm  33249  totprobd  33413  dstrvprob  33458  ballotlem2  33475  requad2  46277  scmsuppss  47001  lincvalsc0  47055  linc0scn0  47057  lincdifsn  47058  linc1  47059  lincsum  47063  lincscm  47064  lcoss  47070  lincext3  47090  islindeps2  47117  itscnhlinecirc02p  47424
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