Theorem atsseq

Description: Two atoms in a subset relationship are equal. (Contributed by NM, 26-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	atsseq	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⊆ 𝐵 ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		atne0	⊢ ( 𝐴 ∈ HAtoms → 𝐴 ≠ 0ℋ )
2	1	ad2antrr	⊢ ( ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 ≠ 0ℋ )
3		atelch	⊢ ( 𝐴 ∈ HAtoms → 𝐴 ∈ Cℋ )
4		atss	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐴 = 0ℋ ) ) )
5	3 4	sylan	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⊆ 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐴 = 0ℋ ) ) )
6	5	imp	⊢ ( ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 = 𝐵 ∨ 𝐴 = 0ℋ ) )
7	6	ord	⊢ ( ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) ∧ 𝐴 ⊆ 𝐵 ) → ( ¬ 𝐴 = 𝐵 → 𝐴 = 0ℋ ) )
8	7	necon1ad	⊢ ( ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐴 ≠ 0ℋ → 𝐴 = 𝐵 ) )
9	2 8	mpd	⊢ ( ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) ∧ 𝐴 ⊆ 𝐵 ) → 𝐴 = 𝐵 )
10	9	ex	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⊆ 𝐵 → 𝐴 = 𝐵 ) )
11		eqimss	⊢ ( 𝐴 = 𝐵 → 𝐴 ⊆ 𝐵 )
12	10 11	impbid1	⊢ ( ( 𝐴 ∈ HAtoms ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 ⊆ 𝐵 ↔ 𝐴 = 𝐵 ) )