Theorem atssch 32950

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 32945	. 2 ⊢ HAtoms = {𝑥 ∈ Cℋ ∣ 0ℋ ⋖ℋ 𝑥}
2	1	ssrab3 4007	1 ⊢ HAtoms ⊆ Cℋ

This proof depends on syntax axioms:   ⊆ wss 3895   class class class wbr 5102   C_ℋ cch 31536  0_ℋc0h 31542   ⋖_ℋ ccv 31571  HAtomscat 31572

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-ss 3896  df-at 32945

This theorem is used by:  atelch  32951  shatomistici  32968  hatomistici  32969  chpssati  32970