Theorem chnlei

Description: Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 5-Jun-2004) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		ch0le.1	⊢ 𝐴 ∈ Cℋ
2		chjcl.2	⊢ 𝐵 ∈ Cℋ
3	1 2	chub1i	⊢ 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 )
4	3	biantrur	⊢ ( ¬ 𝐴 = ( 𝐴 ∨ℋ 𝐵 ) ↔ ( 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ∧ ¬ 𝐴 = ( 𝐴 ∨ℋ 𝐵 ) ) )
5	2 1	chlejb1i	⊢ ( 𝐵 ⊆ 𝐴 ↔ ( 𝐵 ∨ℋ 𝐴 ) = 𝐴 )
6		eqcom	⊢ ( ( 𝐵 ∨ℋ 𝐴 ) = 𝐴 ↔ 𝐴 = ( 𝐵 ∨ℋ 𝐴 ) )
7	2 1	chjcomi	⊢ ( 𝐵 ∨ℋ 𝐴 ) = ( 𝐴 ∨ℋ 𝐵 )
8	7	eqeq2i	⊢ ( 𝐴 = ( 𝐵 ∨ℋ 𝐴 ) ↔ 𝐴 = ( 𝐴 ∨ℋ 𝐵 ) )
9	5 6 8	3bitri	⊢ ( 𝐵 ⊆ 𝐴 ↔ 𝐴 = ( 𝐴 ∨ℋ 𝐵 ) )
10	9	notbii	⊢ ( ¬ 𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 = ( 𝐴 ∨ℋ 𝐵 ) )
11		dfpss2	⊢ ( 𝐴 ⊊ ( 𝐴 ∨ℋ 𝐵 ) ↔ ( 𝐴 ⊆ ( 𝐴 ∨ℋ 𝐵 ) ∧ ¬ 𝐴 = ( 𝐴 ∨ℋ 𝐵 ) ) )
12	4 10 11	3bitr4i	⊢ ( ¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ ( 𝐴 ∨ℋ 𝐵 ) )