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Theorem elpwi 3599
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 3598 . 2 (𝐴 ∈ 𝒫 𝐵 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
21ibi 176 1 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2160  wss 3144  𝒫 cpw 3590
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592
This theorem is referenced by:  elpwid  3601  elelpwi  3602  elpw2g  4171  eldifpw  4492  iunpw  4495  f1opw2  6095  pw1dc1  6931  fi0  6992  pw1on  7243  lspf  13666  cnntr  14122
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