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Theorem elelpwi 4330
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 4327 . . 3 (𝐵 ∈ 𝒫 𝐶𝐵𝐶)
21sseld 3762 . 2 (𝐵 ∈ 𝒫 𝐶 → (𝐴𝐵𝐴𝐶))
32impcom 396 1 ((𝐴𝐵𝐵 ∈ 𝒫 𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 2155  𝒫 cpw 4317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-in 3741  df-ss 3748  df-pw 4319
This theorem is referenced by:  unipw  5076  axdc2lem  9527  axdc3lem4  9532  homarel  16965  txdis  21729  uhgredgrnv  26316  fpwrelmap  29978  insiga  30668  measinblem  30751  ddemeas  30767  imambfm  30792  totprobd  30957  dstrvprob  31002  ballotlem2  31019  scmsuppss  42846  lincvalsc0  42903  linc0scn0  42905  lincdifsn  42906  linc1  42907  lincsum  42911  lincscm  42912  lcoss  42918  lincext3  42938  islindeps2  42965
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