Theorem atssch 29511

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 29506	. 2 ⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥}
2		ssrab2 3828	. 2 ⊢ {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥} ⊆ Cℋ
3	1, 2	eqsstri 3776	1 ⊢ HAtoms ⊆ Cℋ

Syntax hints:  {crab 3054   ⊆ wss 3715   class class class wbr 4804   C_ℋ cch 28095  0_ℋc0h 28101   ⋖_ℋ ccv 28130  HAtomscat 28131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-in 3722  df-ss 3729  df-at 29506

This theorem is referenced by:  atelch  29512  shatomistici  29529  hatomistici  29530  chpssati  29531