Theorem elatcv0 32943

Description: A Hilbert lattice element is an atom iff it covers the zero subspace. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
elatcv0	⊢ (𝐴 ∈ Cℋ → (𝐴 ∈ HAtoms ↔ 0ℋ ⋖ℋ 𝐴))

Proof of Theorem elatcv0

Step	Hyp	Ref	Expression
1		ela 32941	. 2 ⊢ (𝐴 ∈ HAtoms ↔ (𝐴 ∈ Cℋ ∧ 0ℋ ⋖ℋ 𝐴))
2	1	baib 545	1 ⊢ (𝐴 ∈ Cℋ → (𝐴 ∈ HAtoms ↔ 0ℋ ⋖ℋ 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∈ wcel 2145   class class class wbr 5103   C_ℋ cch 31531  0_ℋc0h 31537   ⋖_ℋ ccv 31566  HAtomscat 31567

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 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-at 32940

This theorem is used by: (None)