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Theorem elelpwi 4613
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 4610 . . 3 (𝐵 ∈ 𝒫 𝐶𝐵𝐶)
21sseld 3982 . 2 (𝐵 ∈ 𝒫 𝐶 → (𝐴𝐵𝐴𝐶))
32impcom 409 1 ((𝐴𝐵𝐵 ∈ 𝒫 𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  𝒫 cpw 4603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-pw 4605
This theorem is referenced by:  unipw  5451  axdc2lem  10443  axdc3lem4  10448  homarel  17986  txdis  23136  uhgredgrnv  28390  fpwrelmap  31958  insiga  33135  measinblem  33218  ddemeas  33234  imambfm  33261  totprobd  33425  dstrvprob  33470  ballotlem2  33487  requad2  46291  scmsuppss  47048  lincvalsc0  47102  linc0scn0  47104  lincdifsn  47105  linc1  47106  lincsum  47110  lincscm  47111  lcoss  47117  lincext3  47137  islindeps2  47164  itscnhlinecirc02p  47471
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