Theorem dochsordN

Description: Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	doch11.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	doch11.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
	doch11.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	doch11.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	doch11.x	⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 )
	doch11.y	⊢ ( 𝜑 → 𝑌 ∈ ran 𝐼 )
Assertion	dochsordN	⊢ ( 𝜑 → ( 𝑋 ⊊ 𝑌 ↔ ( ⊥ ‘ 𝑌 ) ⊊ ( ⊥ ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		doch11.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		doch11.i	⊢ 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
3		doch11.o	⊢ ⊥ = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
4		doch11.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
5		doch11.x	⊢ ( 𝜑 → 𝑋 ∈ ran 𝐼 )
6		doch11.y	⊢ ( 𝜑 → 𝑌 ∈ ran 𝐼 )
7	1 2 3 4 5 6	dochord	⊢ ( 𝜑 → ( 𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ) )
8	1 2 3 4 6 5	doch11	⊢ ( 𝜑 → ( ( ⊥ ‘ 𝑌 ) = ( ⊥ ‘ 𝑋 ) ↔ 𝑌 = 𝑋 ) )
9		eqcom	⊢ ( 𝑌 = 𝑋 ↔ 𝑋 = 𝑌 )
10	8 9	bitr2di	⊢ ( 𝜑 → ( 𝑋 = 𝑌 ↔ ( ⊥ ‘ 𝑌 ) = ( ⊥ ‘ 𝑋 ) ) )
11	10	necon3bid	⊢ ( 𝜑 → ( 𝑋 ≠ 𝑌 ↔ ( ⊥ ‘ 𝑌 ) ≠ ( ⊥ ‘ 𝑋 ) ) )
12	7 11	anbi12d	⊢ ( 𝜑 → ( ( 𝑋 ⊆ 𝑌 ∧ 𝑋 ≠ 𝑌 ) ↔ ( ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ≠ ( ⊥ ‘ 𝑋 ) ) ) )
13		df-pss	⊢ ( 𝑋 ⊊ 𝑌 ↔ ( 𝑋 ⊆ 𝑌 ∧ 𝑋 ≠ 𝑌 ) )
14		df-pss	⊢ ( ( ⊥ ‘ 𝑌 ) ⊊ ( ⊥ ‘ 𝑋 ) ↔ ( ( ⊥ ‘ 𝑌 ) ⊆ ( ⊥ ‘ 𝑋 ) ∧ ( ⊥ ‘ 𝑌 ) ≠ ( ⊥ ‘ 𝑋 ) ) )
15	12 13 14	3bitr4g	⊢ ( 𝜑 → ( 𝑋 ⊊ 𝑌 ↔ ( ⊥ ‘ 𝑌 ) ⊊ ( ⊥ ‘ 𝑋 ) ) )