Theorem elelpwi 4560

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 4557	. . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 → 𝐵 ⊆ 𝐶)
2	1	sseld 3933	. 2 ⊢ (𝐵 ∈ 𝒫 𝐶 → (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐶))
3	2	impcom 407	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2111  𝒫 cpw 4550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ss 3919  df-pw 4552

This theorem is referenced by:  unipw  5391  axdc2lem  10336  axdc3lem4  10341  homarel  17940  txdis  23545  uhgredgrnv  29106  fpwrelmap  32711  insiga  34145  measinblem  34228  ddemeas  34244  imambfm  34270  totprobd  34434  dstrvprob  34480  ballotlem2  34497  requad2  47653  scmsuppss  48401  lincvalsc0  48452  linc0scn0  48454  lincdifsn  48455  linc1  48456  lincsum  48460  lincscm  48461  lcoss  48467  lincext3  48487  islindeps2  48514  itscnhlinecirc02p  48816