This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 dfss2 ⊢ ( 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 ) )