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Metamath Proof Explorer

Theorem opltcon2b

Description: Contraposition law for strict ordering in orthoposets. ( chsscon2 analog.) (Contributed by NM, 5-Nov-2011)

Ref Expression
Hypotheses opltcon3.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
opltcon3.s ⊢ < = ( lt ‘ 𝐾 )
opltcon3.o ⊢ ⊥ = ( oc ‘ 𝐾 )
Assertion opltcon2b ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < ( ⊥ ‘ 𝑌 ) ↔ 𝑌 < ( ⊥ ‘ 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 opltcon3.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 opltcon3.s ⊢ < = ( lt ‘ 𝐾 )
3 opltcon3.o ⊢ ⊥ = ( oc ‘ 𝐾 )
4 1 3 opoccl ⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 )
5 4 3adant2 ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ 𝑌 ) ∈ 𝐵 )
6 1 2 3 opltcon3b ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘ 𝑌 ) ∈ 𝐵 ) → ( 𝑋 < ( ⊥ ‘ 𝑌 ) ↔ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) < ( ⊥ ‘ 𝑋 ) ) )
7 5 6 syld3an3 ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < ( ⊥ ‘ 𝑌 ) ↔ ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) < ( ⊥ ‘ 𝑋 ) ) )
8 1 3 opococ ⊢ ( ( 𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 )
9 8 3adant2 ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) = 𝑌 )
10 9 breq1d ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ⊥ ‘ ( ⊥ ‘ 𝑌 ) ) < ( ⊥ ‘ 𝑋 ) ↔ 𝑌 < ( ⊥ ‘ 𝑋 ) ) )
11 7 10 bitrd ⊢ ( ( 𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < ( ⊥ ‘ 𝑌 ) ↔ 𝑌 < ( ⊥ ‘ 𝑋 ) ) )