Metamath Proof Explorer


Theorem opltcon1b

Description: Contraposition law for strict ordering in orthoposets. ( chpsscon1 analog.) (Contributed by NM, 5-Nov-2011)

Ref Expression
Hypotheses opltcon3.b 𝐵 = ( Base ‘ 𝐾 )
opltcon3.s < = ( lt ‘ 𝐾 )
opltcon3.o = ( oc ‘ 𝐾 )
Assertion opltcon1b ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 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 ∧ 𝑋𝐵 ) → ( 𝑋 ) ∈ 𝐵 )
5 4 3adant3 ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 ) ∈ 𝐵 )
6 1 2 3 opltcon3b ( ( 𝐾 ∈ OP ∧ ( 𝑋 ) ∈ 𝐵𝑌𝐵 ) → ( ( 𝑋 ) < 𝑌 ↔ ( 𝑌 ) < ( ‘ ( 𝑋 ) ) ) )
7 5 6 syld3an2 ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 𝑋 ) < 𝑌 ↔ ( 𝑌 ) < ( ‘ ( 𝑋 ) ) ) )
8 1 3 opococ ( ( 𝐾 ∈ OP ∧ 𝑋𝐵 ) → ( ‘ ( 𝑋 ) ) = 𝑋 )
9 8 3adant3 ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵 ) → ( ‘ ( 𝑋 ) ) = 𝑋 )
10 9 breq2d ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 𝑌 ) < ( ‘ ( 𝑋 ) ) ↔ ( 𝑌 ) < 𝑋 ) )
11 7 10 bitrd ( ( 𝐾 ∈ OP ∧ 𝑋𝐵𝑌𝐵 ) → ( ( 𝑋 ) < 𝑌 ↔ ( 𝑌 ) < 𝑋 ) )