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Theorem elelpwi 4574
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 4571 . . 3 (𝐵 ∈ 𝒫 𝐶𝐵𝐶)
21sseld 3947 . 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 4564
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 3449  df-in 3921  df-ss 3931  df-pw 4566
This theorem is referenced by:  unipw  5411  axdc2lem  10392  axdc3lem4  10397  homarel  17930  txdis  23006  uhgredgrnv  28130  fpwrelmap  31704  insiga  32800  measinblem  32883  ddemeas  32899  imambfm  32926  totprobd  33090  dstrvprob  33135  ballotlem2  33152  requad2  45905  scmsuppss  46538  lincvalsc0  46592  linc0scn0  46594  lincdifsn  46595  linc1  46596  lincsum  46600  lincscm  46601  lcoss  46607  lincext3  46627  islindeps2  46654  itscnhlinecirc02p  46961
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