Theorem elelpwi 4577

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 4574	. . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 → 𝐵 ⊆ 𝐶)
2	1	sseld 3939	. 2 ⊢ (𝐵 ∈ 𝒫 𝐶 → (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐶))
3	2	impcom 413	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  𝒫 cpw 4567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ss 3925  df-pw 4569

This theorem is used by:  unipw  5436  axdc2lem  10450  axdc3lem4  10455  homarel  18118  txdis  23826  uhgredgrnv  29517  fpwrelmap  33115  insiga  34559  measinblem  34642  ddemeas  34658  imambfm  34684  totprobd  34848  dstrvprob  34894  ballotlem2  34911  requad2  48429  scmsuppss  49192  lincvalsc0  49242  linc0scn0  49244  lincdifsn  49245  linc1  49246  lincsum  49250  lincscm  49251  lcoss  49257  lincext3  49277  islindeps2  49304  itscnhlinecirc02p  49606