Theorem elelpwi 4573

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 4570	. . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 → 𝐵 ⊆ 𝐶)
2	1	sseld 3945	. 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 2109  𝒫 cpw 4563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ss 3931  df-pw 4565

This theorem is referenced by:  unipw  5410  axdc2lem  10401  axdc3lem4  10406  homarel  17998  txdis  23519  uhgredgrnv  29057  fpwrelmap  32656  insiga  34127  measinblem  34210  ddemeas  34226  imambfm  34253  totprobd  34417  dstrvprob  34463  ballotlem2  34480  requad2  47624  scmsuppss  48359  lincvalsc0  48410  linc0scn0  48412  lincdifsn  48413  linc1  48414  lincsum  48418  lincscm  48419  lcoss  48425  lincext3  48445  islindeps2  48472  itscnhlinecirc02p  48774