Theorem chlejb1i

Description: Hilbert lattice ordering in terms of join. (Contributed by NM, 15-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ch0le.1	⊢ 𝐴 ∈ Cℋ
	chjcl.2	⊢ 𝐵 ∈ Cℋ
Assertion	chlejb1i	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∨ℋ 𝐵 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	⊢ 𝐴 ∈ Cℋ
2		chjcl.2	⊢ 𝐵 ∈ Cℋ
3		ssid	⊢ 𝐵 ⊆ 𝐵
4	1 2 2	chlubii	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐵 ) → ( 𝐴 ∨ℋ 𝐵 ) ⊆ 𝐵 )
5	3 4	mpan2	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ 𝐵 ) ⊆ 𝐵 )
6	2 1	chub2i	⊢ 𝐵 ⊆ ( 𝐴 ∨ℋ 𝐵 )
7	5 6	jctir	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝐴 ∨ℋ 𝐵 ) ⊆ 𝐵 ∧ 𝐵 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) )
8		eqss	⊢ ( ( 𝐴 ∨ℋ 𝐵 ) = 𝐵 ↔ ( ( 𝐴 ∨ℋ 𝐵 ) ⊆ 𝐵 ∧ 𝐵 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ) )
9	7 8	sylibr	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∨ℋ 𝐵 ) = 𝐵 )
10	1 2	chub1i	⊢ 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 )
11		eqimss	⊢ ( ( 𝐴 ∨ℋ 𝐵 ) = 𝐵 → ( 𝐴 ∨ℋ 𝐵 ) ⊆ 𝐵 )
12	10 11	sstrid	⊢ ( ( 𝐴 ∨ℋ 𝐵 ) = 𝐵 → 𝐴 ⊆ 𝐵 )
13	9 12	impbii	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∨ℋ 𝐵 ) = 𝐵 )