Theorem atne0

Description: An atom is not the Hilbert lattice zero. (Contributed by NM, 13-Aug-2002) (New usage is discouraged.)

	Ref	Expression
Assertion	atne0	⊢ ( 𝐴 ∈ HAtoms → 𝐴 ≠ 0ℋ )

Proof

Step	Hyp	Ref	Expression
1		elat2	⊢ ( 𝐴 ∈ HAtoms ↔ ( 𝐴 ∈ Cℋ ∧ ( 𝐴 ≠ 0ℋ ∧ ∀ 𝑥 ∈ Cℋ ( 𝑥 ⊆ 𝐴 → ( 𝑥 = 𝐴 ∨ 𝑥 = 0ℋ ) ) ) ) )
2		simprl	⊢ ( ( 𝐴 ∈ Cℋ ∧ ( 𝐴 ≠ 0ℋ ∧ ∀ 𝑥 ∈ Cℋ ( 𝑥 ⊆ 𝐴 → ( 𝑥 = 𝐴 ∨ 𝑥 = 0ℋ ) ) ) ) → 𝐴 ≠ 0ℋ )
3	1 2	sylbi	⊢ ( 𝐴 ∈ HAtoms → 𝐴 ≠ 0ℋ )