Theorem elpwinss 46282

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 𝐵 ∩ 𝐶) → 𝐴 ⊆ 𝐵)

Proof of Theorem elpwinss

Step	Hyp	Ref	Expression
1		elinel1 4101	. 2 ⊢ (𝐴 ∈ (𝒫 𝐵 ∩ 𝐶) → 𝐴 ∈ 𝒫 𝐵)
2	1	elpwid 4569	1 ⊢ (𝐴 ∈ (𝒫 𝐵 ∩ 𝐶) → 𝐴 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-in 4094  df-pw 4562

This theorem is used by:  sge0z  47600  sge0revalmpt  47603  sge0f1o  47607  sge0rnbnd  47618  sge0pnffigt  47621  sge0lefi  47623  sge0ltfirp  47625  sge0gerpmpt  47627  sge0le  47632  sge0ltfirpmpt  47633  sge0iunmptlemre  47640  sge0rpcpnf  47646  sge0lefimpt  47648  sge0ltfirpmpt2  47651  sge0isum  47652  sge0xaddlem1  47658  sge0xaddlem2  47659  sge0pnffigtmpt  47665  sge0pnffsumgt  47667  sge0gtfsumgt  47668  sge0uzfsumgt  47669  sge0seq  47671  sge0reuz  47672  omeiunltfirp  47744  carageniuncllem2  47747  caratheodorylem2  47752