Theorem elpwinss 39216

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 3799	. 2 ⊢ (𝐴 ∈ (𝒫 𝐵 ∩ 𝐶) → 𝐴 ∈ 𝒫 𝐵)
2	1	elpwid 4170	1 ⊢ (𝐴 ∈ (𝒫 𝐵 ∩ 𝐶) → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 1990   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160

This theorem is referenced by:  sge0z  40592  sge0revalmpt  40595  sge0f1o  40599  sge0rnbnd  40610  sge0pnffigt  40613  sge0lefi  40615  sge0ltfirp  40617  sge0gerpmpt  40619  sge0le  40624  sge0ltfirpmpt  40625  sge0iunmptlemre  40632  sge0rpcpnf  40638  sge0lefimpt  40640  sge0ltfirpmpt2  40643  sge0isum  40644  sge0xaddlem1  40650  sge0xaddlem2  40651  sge0pnffigtmpt  40657  sge0pnffsumgt  40659  sge0gtfsumgt  40660  sge0uzfsumgt  40661  sge0seq  40663  sge0reuz  40664  omeiunltfirp  40733  carageniuncllem2  40736  caratheodorylem2  40741