Theorem atssch 32732

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 32727	. 2 ⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥}
2	1	ssrab3 4039	1 ⊢ HAtoms ⊆ Cℋ

This proof depends on syntax axioms:   ⊆ wss 3908   class class class wbr 5114   C_ℋ cch 31318  0_ℋc0h 31324   ⋖_ℋ ccv 31353  HAtomscat 31354

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-ss 3925  df-at 32727

This theorem is used by:  atelch  32733  shatomistici  32750  hatomistici  32751  chpssati  32752