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Theorem elelpwi 4556
 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 4553 . . 3 (𝐵 ∈ 𝒫 𝐶𝐵𝐶)
21sseld 3969 . 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 4541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-in 3946  df-ss 3955  df-pw 4543 This theorem is referenced by:  unipw  5338  axdc2lem  9862  axdc3lem4  9867  homarel  17288  txdis  22156  uhgredgrnv  26830  fpwrelmap  30383  insiga  31283  measinblem  31366  ddemeas  31382  imambfm  31407  totprobd  31571  dstrvprob  31616  ballotlem2  31633  requad2  43617  scmsuppss  44249  lincvalsc0  44305  linc0scn0  44307  lincdifsn  44308  linc1  44309  lincsum  44313  lincscm  44314  lcoss  44320  lincext3  44340  islindeps2  44367  itscnhlinecirc02p  44601
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