Theorem atssch 32414

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 32409	. 2 ⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥}
2	1	ssrab3 4022	1 ⊢ HAtoms ⊆ Cℋ

Syntax hints:   ⊆ wss 3889   class class class wbr 5085   C_ℋ cch 31000  0_ℋc0h 31006   ⋖_ℋ ccv 31035  HAtomscat 31036

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-ss 3906  df-at 32409

This theorem is referenced by:  atelch  32415  shatomistici  32432  hatomistici  32433  chpssati  32434