Theorem atssma

Description: The meet with an atom's superset is the atom. (Contributed by NM, 12-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	atssma	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) ∈ HAtoms ) )

Proof

Step	Hyp	Ref	Expression
1		df-ss	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) = 𝐴 )
2	1	biimpi	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ 𝐵 ) = 𝐴 )
3	2	eleq1d	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝐴 ∩ 𝐵 ) ∈ HAtoms ↔ 𝐴 ∈ HAtoms ) )
4	3	biimprcd	⊢ ( 𝐴 ∈ HAtoms → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ 𝐵 ) ∈ HAtoms ) )
5	4	adantr	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ 𝐵 ) ∈ HAtoms ) )
6		incom	⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐴 )
7	6	eleq1i	⊢ ( ( 𝐴 ∩ 𝐵 ) ∈ HAtoms ↔ ( 𝐵 ∩ 𝐴 ) ∈ HAtoms )
8		atne0	⊢ ( ( 𝐵 ∩ 𝐴 ) ∈ HAtoms → ( 𝐵 ∩ 𝐴 ) ≠ 0ℋ )
9	8	neneqd	⊢ ( ( 𝐵 ∩ 𝐴 ) ∈ HAtoms → ¬ ( 𝐵 ∩ 𝐴 ) = 0ℋ )
10	7 9	sylbi	⊢ ( ( 𝐴 ∩ 𝐵 ) ∈ HAtoms → ¬ ( 𝐵 ∩ 𝐴 ) = 0ℋ )
11		atnssm0	⊢ ( ( 𝐵 ∈ Cℋ ∧ 𝐴 ∈ HAtoms ) → ( ¬ 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐴 ) = 0ℋ ) )
12	11	ancoms	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → ( ¬ 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐴 ) = 0ℋ ) )
13	12	biimpd	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → ( ¬ 𝐴 ⊆ 𝐵 → ( 𝐵 ∩ 𝐴 ) = 0ℋ ) )
14	13	con1d	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → ( ¬ ( 𝐵 ∩ 𝐴 ) = 0ℋ → 𝐴 ⊆ 𝐵 ) )
15	10 14	syl5	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → ( ( 𝐴 ∩ 𝐵 ) ∈ HAtoms → 𝐴 ⊆ 𝐵 ) )
16	5 15	impbid	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ Cℋ ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) ∈ HAtoms ) )