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Metamath Proof Explorer

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 ) ) )