Theorem polcon2bN

Description: Contraposition law for polarity. (Contributed by NM, 23-Mar-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	2polss.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	2polss.p	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
Assertion	polcon2bN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ↔ 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		2polss.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2		2polss.p	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
3		simpl1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝐾 ∈ HL )
4		simpl3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝑌 ⊆ 𝐴 )
5		simpr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) )
6	1 2	polcon2N	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) )
7	3 4 5 6	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ) → 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) )
8		simpl1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) ) → 𝐾 ∈ HL )
9		simpl2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) ) → 𝑋 ⊆ 𝐴 )
10		simpr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) ) → 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) )
11	1 2	polcon2N	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) ) → 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) )
12	8 9 10 11	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) ∧ 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) ) → 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) )
13	7 12	impbida	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ) → ( 𝑋 ⊆ ( ⊥ ‘ 𝑌 ) ↔ 𝑌 ⊆ ( ⊥ ‘ 𝑋 ) ) )