Theorem atssch 31851

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 31846	. 2 ⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥}
2	1	ssrab3 4080	1 ⊢ HAtoms ⊆ Cℋ

Syntax hints:   ⊆ wss 3948   class class class wbr 5148   C_ℋ cch 30437  0_ℋc0h 30443   ⋖_ℋ ccv 30472  HAtomscat 30473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-in 3955  df-ss 3965  df-at 31846

This theorem is referenced by:  atelch  31852  shatomistici  31869  hatomistici  31870  chpssati  31871