Theorem atssch 29202

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 29197	. 2 ⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥}
2		ssrab2 3687	. 2 ⊢ {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥} ⊆ Cℋ
3	1, 2	eqsstri 3635	1 ⊢ HAtoms ⊆ Cℋ

Syntax hints:  {crab 2916   ⊆ wss 3574   class class class wbr 4653   C_ℋ cch 27786  0_ℋc0h 27792   ⋖_ℋ ccv 27821  HAtomscat 27822

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-rab 2921  df-in 3581  df-ss 3588  df-at 29197

This theorem is referenced by:  atelch  29203  shatomistici  29220  hatomistici  29221  chpssati  29222