Theorem elelpwi 4612

Description: If 𝐴 belongs to a part of 𝐶, then 𝐴 belongs to 𝐶. (Contributed by FL, 3-Aug-2009.)

Assertion

Ref	Expression
elelpwi	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶)

Proof of Theorem elelpwi

Step	Hyp	Ref	Expression
1		elpwi 4609	. . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 → 𝐵 ⊆ 𝐶)
2	1	sseld 3981	. 2 ⊢ (𝐵 ∈ 𝒫 𝐶 → (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐶))
3	2	impcom 408	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  𝒫 cpw 4602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3955  df-ss 3965  df-pw 4604

This theorem is referenced by:  unipw  5450  axdc2lem  10442  axdc3lem4  10447  homarel  17985  txdis  23135  uhgredgrnv  28387  fpwrelmap  31953  insiga  33130  measinblem  33213  ddemeas  33229  imambfm  33256  totprobd  33420  dstrvprob  33465  ballotlem2  33482  requad2  46281  scmsuppss  47038  lincvalsc0  47092  linc0scn0  47094  lincdifsn  47095  linc1  47096  lincsum  47100  lincscm  47101  lcoss  47107  lincext3  47127  islindeps2  47154  itscnhlinecirc02p  47461