Theorem elelpwi 3641

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 3638	. . 3 ⊢ (𝐵 ∈ 𝒫 𝐶 → 𝐵 ⊆ 𝐶)
2	1	sseld 3203	. 2 ⊢ (𝐵 ∈ 𝒫 𝐶 → (𝐴 ∈ 𝐵 → 𝐴 ∈ 𝐶))
3	2	impcom 125	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝒫 𝐶) → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2180  𝒫 cpw 3629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-in 3183  df-ss 3190  df-pw 3631

This theorem is referenced by:  unipw  4282  txdis  14916  uhgredgrnv  15901