Theorem elpwi 3568

Description: Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)

Assertion

Ref	Expression
elpwi	⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)

Proof of Theorem elpwi

Step	Hyp	Ref	Expression
1		elpwg 3567	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))
2	1	ibi 175	1 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2136   ⊆ wss 3116  𝒫 cpw 3559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561

This theorem is referenced by:  elpwid  3570  elelpwi  3571  elpw2g  4135  eldifpw  4455  iunpw  4458  f1opw2  6044  pw1dc1  6879  fi0  6940  pw1on  7182  cnntr  12865