Theorem elpwinss 38241

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 3761	. 2 ⊢ (𝐴 ∈ (𝒫 𝐵 ∩ 𝐶) → 𝐴 ∈ 𝒫 𝐵)
2		elpwi 4117	. 2 ⊢ (𝐴 ∈ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ (𝒫 𝐵 ∩ 𝐶) → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110

This theorem is referenced by:  sge0z  39268  sge0revalmpt  39271  sge0f1o  39275  sge0rnbnd  39286  sge0pnffigt  39289  sge0lefi  39291  sge0ltfirp  39293  sge0gerpmpt  39295  sge0le  39300  sge0ltfirpmpt  39301  sge0iunmptlemre  39308  sge0rpcpnf  39314  sge0lefimpt  39316  sge0ltfirpmpt2  39319  sge0isum  39320  sge0xaddlem1  39326  sge0xaddlem2  39327  sge0pnffigtmpt  39333  sge0pnffsumgt  39335  sge0gtfsumgt  39336  sge0uzfsumgt  39337  sge0seq  39339  sge0reuz  39340  omeiunltfirp  39409  carageniuncllem2  39412  caratheodorylem2  39417