Theorem atssch 5741

Description: Atoms are a subset of the Hilbert lattice.

Assertion

Ref	Expression
atssch	⊢ Atoms ⊆ Cℋ

Proof of Theorem atssch

Step	Hyp	Ref	Expression
1		df-at 5737	. 2 ⊢ Atoms = {x ∈ Cℋ ∣0ℋ ⋖ x}
2		ssrab 1556	. 2 ⊢ {x ∈ Cℋ ∣0ℋ ⋖ x} ⊆ Cℋ
3	1, 2	eqsstr 1530	1 ⊢ Atoms ⊆ Cℋ

Syntax hints:  {crab 1204   ⊆ wss 1487   class class class wbr 2054   C_ℋ cch 4968  0_ℋc0h 4974  Atomscat 4980   ⋖ ccv 4981

This theorem is referenced by:  atelch 5742  hatomistic 5755  chpssat 5756

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208  df-in 1491  df-ss 1492  df-at 5737

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