Theorem elelpwi 4564

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 4561	. . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 → 𝐵 ⊆ 𝐶)
2	1	sseld 3935	. 2 ⊢ (𝐵 ∈ 𝒫 𝐶 → (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐶))
3	2	impcom 411	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2141  𝒫 cpw 4554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ss 3921  df-pw 4556

This theorem is referenced by:  unipw  5416  axdc2lem  10402  axdc3lem4  10407  homarel  18052  txdis  23672  uhgredgrnv  29277  fpwrelmap  32885  insiga  34395  measinblem  34478  ddemeas  34494  imambfm  34520  totprobd  34684  dstrvprob  34730  ballotlem2  34747  requad2  48209  scmsuppss  48957  lincvalsc0  49007  linc0scn0  49009  lincdifsn  49010  linc1  49011  lincsum  49015  lincscm  49016  lcoss  49022  lincext3  49042  islindeps2  49069  itscnhlinecirc02p  49371