Theorem atssch 28586

Description: Atoms are a subset of the Hilbert lattice. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)

Assertion

Ref	Expression
atssch	⊢ HAtoms ⊆ Cℋ

Proof of Theorem atssch

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-at 28581	. 2 ⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥}
2		ssrab2 3650	. 2 ⊢ {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥} ⊆ Cℋ
3	1, 2	eqsstri 3598	1 ⊢ HAtoms ⊆ Cℋ

Syntax hints:  {crab 2900   ⊆ wss 3540   class class class wbr 4583   C_ℋ cch 27170  0_ℋc0h 27176   ⋖_ℋ ccv 27205  HAtomscat 27206

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-rab 2905  df-in 3547  df-ss 3554  df-at 28581

This theorem is referenced by:  atelch  28587  shatomistici  28604  hatomistici  28605  chpssati  28606