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Theorem elpwinss 39733
Description: An element of the powerset of 𝐵 intersected with anything, is a subset of 𝐵. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
elpwinss (𝐴 ∈ (𝒫 𝐵𝐶) → 𝐴𝐵)

Proof of Theorem elpwinss
StepHypRef Expression
1 elinel1 3942 . 2 (𝐴 ∈ (𝒫 𝐵𝐶) → 𝐴 ∈ 𝒫 𝐵)
21elpwid 4314 1 (𝐴 ∈ (𝒫 𝐵𝐶) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139  cin 3714  wss 3715  𝒫 cpw 4302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729  df-pw 4304
This theorem is referenced by:  sge0z  41113  sge0revalmpt  41116  sge0f1o  41120  sge0rnbnd  41131  sge0pnffigt  41134  sge0lefi  41136  sge0ltfirp  41138  sge0gerpmpt  41140  sge0le  41145  sge0ltfirpmpt  41146  sge0iunmptlemre  41153  sge0rpcpnf  41159  sge0lefimpt  41161  sge0ltfirpmpt2  41164  sge0isum  41165  sge0xaddlem1  41171  sge0xaddlem2  41172  sge0pnffigtmpt  41178  sge0pnffsumgt  41180  sge0gtfsumgt  41181  sge0uzfsumgt  41182  sge0seq  41184  sge0reuz  41185  omeiunltfirp  41257  carageniuncllem2  41260  caratheodorylem2  41265
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