Theorem atssch 32978

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 32973	. 2 ⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥}
2	1	ssrab3 4010	1 ⊢ HAtoms ⊆ Cℋ

This proof depends on syntax axioms:   ⊆ wss 3900   class class class wbr 5106   C_ℋ cch 31564  0_ℋc0h 31570   ⋖_ℋ ccv 31599  HAtomscat 31600

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-ss 3901  df-at 32973

This theorem is used by:  atelch  32979  shatomistici  32996  hatomistici  32997  chpssati  32998