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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 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ( ¬ 𝐵 ⊆ 𝐴 ↔ 𝐴 ⊊ ( 𝐴 ∨ℋ 𝐵 ) ) )

Proof

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