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  10445  axdc3lem4  10450  homarel  17990  txdis  23356  uhgredgrnv  28645  fpwrelmap  32213  insiga  33421  measinblem  33504  ddemeas  33520  imambfm  33547  totprobd  33711  dstrvprob  33756  ballotlem2  33773  requad2  46590  scmsuppss  47137  lincvalsc0  47190  linc0scn0  47192  lincdifsn  47193  linc1  47194  lincsum  47198  lincscm  47199  lcoss  47205  lincext3  47225  islindeps2  47252  itscnhlinecirc02p  47559