Metamath Proof Explorer

Theorem chnle

Description: Equivalent expressions for "not less than" in the Hilbert lattice. (Contributed by NM, 9-Jun-2004) (New usage is discouraged.)

		Ref	Expression
	Assertion	chnle	⊢ ( ( 𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ ( 𝐴 ∨_ℋ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sseq2	⊢ ( 𝐴 = if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) → ( 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ) )
2	1	notbid	⊢ ( 𝐴 = if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ ¬ 𝐵 ⊆ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ) )
3		id	⊢ ( 𝐴 = if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) → 𝐴 = if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) )
4		oveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) → ( 𝐴 ∨_ℋ 𝐵 ) = ( if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ∨_ℋ 𝐵 ) )
5	3 4	psseq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) → ( 𝐴 ⊊ ( 𝐴 ∨_ℋ 𝐵 ) ↔ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ⊊ ( if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ∨_ℋ 𝐵 ) ) )
6	2 5	bibi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) → ( ( ¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ ( 𝐴 ∨_ℋ 𝐵 ) ) ↔ ( ¬ 𝐵 ⊆ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ↔ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ⊊ ( if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ∨_ℋ 𝐵 ) ) ) )
7		sseq1	⊢ ( 𝐵 = if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) → ( 𝐵 ⊆ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ↔ if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) ⊆ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ) )
8	7	notbid	⊢ ( 𝐵 = if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) → ( ¬ 𝐵 ⊆ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ↔ ¬ if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) ⊆ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ) )
9		oveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) → ( if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ∨_ℋ 𝐵 ) = ( if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ∨_ℋ if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) ) )
10	9	psseq2d	⊢ ( 𝐵 = if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) → ( if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ⊊ ( if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ∨_ℋ 𝐵 ) ↔ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ⊊ ( if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ∨_ℋ if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) ) ) )
11	8 10	bibi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) → ( ( ¬ 𝐵 ⊆ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ↔ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ⊊ ( if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ∨_ℋ 𝐵 ) ) ↔ ( ¬ if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) ⊆ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ↔ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ⊊ ( if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ∨_ℋ if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) ) ) ) )
12		h0elch	⊢ 0_ℋ ∈ C_ℋ
13	12	elimel	⊢ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ∈ C_ℋ
14	12	elimel	⊢ if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) ∈ C_ℋ
15	13 14	chnlei	⊢ ( ¬ if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) ⊆ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ↔ if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ⊊ ( if ( 𝐴 ∈ C_ℋ , 𝐴 , 0_ℋ ) ∨_ℋ if ( 𝐵 ∈ C_ℋ , 𝐵 , 0_ℋ ) ) )
16	6 11 15	dedth2h	⊢ ( ( 𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ ( 𝐴 ∨_ℋ 𝐵 ) ) )