Theorem chnlen0

Description: A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chnlen0	⊢ ( 𝐵 ∈ Cℋ → ( ¬ 𝐴 ⊆ 𝐵 → ¬ 𝐴 = 0ℋ ) )

Proof

Step	Hyp	Ref	Expression
1		ch0le	⊢ ( 𝐵 ∈ Cℋ → 0ℋ ⊆ 𝐵 )
2		sseq1	⊢ ( 𝐴 = 0ℋ → ( 𝐴 ⊆ 𝐵 ↔ 0ℋ ⊆ 𝐵 ) )
3	1 2	syl5ibrcom	⊢ ( 𝐵 ∈ Cℋ → ( 𝐴 = 0ℋ → 𝐴 ⊆ 𝐵 ) )
4	3	con3d	⊢ ( 𝐵 ∈ Cℋ → ( ¬ 𝐴 ⊆ 𝐵 → ¬ 𝐴 = 0ℋ ) )