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Theorem elpwi 3489
Description: Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
Assertion
Ref Expression
elpwi (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Proof of Theorem elpwi
StepHypRef Expression
1 elpwg 3488 . 2 (𝐴 ∈ 𝒫 𝐵 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
21ibi 175 1 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1465  wss 3041  𝒫 cpw 3480
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482
This theorem is referenced by:  elpwid  3491  elelpwi  3492  elpw2g  4051  eldifpw  4368  iunpw  4371  f1opw2  5944  fi0  6831  cnntr  12321
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