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Theorem elelpwi 4550
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 4547 . . 3 (𝐵 ∈ 𝒫 𝐶𝐵𝐶)
21sseld 3963 . 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 2105  𝒫 cpw 4535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-in 3940  df-ss 3949  df-pw 4537
This theorem is referenced by:  unipw  5333  axdc2lem  9858  axdc3lem4  9863  homarel  17284  txdis  22168  uhgredgrnv  26842  fpwrelmap  30395  insiga  31295  measinblem  31378  ddemeas  31394  imambfm  31419  totprobd  31583  dstrvprob  31628  ballotlem2  31645  requad2  43665  scmsuppss  44348  lincvalsc0  44404  linc0scn0  44406  lincdifsn  44407  linc1  44408  lincsum  44412  lincscm  44413  lcoss  44419  lincext3  44439  islindeps2  44466  itscnhlinecirc02p  44700
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