opltcon2b - Mathbox for Norm Megill

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Theorem opltcon2b 39666

Description: Contraposition law for strict ordering in orthoposets. (chsscon2 31588 analog.) (Contributed by NM, 5-Nov-2011.)

**Hypotheses**
Ref	Expression
opltcon3.b	⊢ 𝐵 = (Base‘𝐾)
opltcon3.s	⊢ < = (lt‘𝐾)
opltcon3.o	⊢ ⊥ = (oc‘𝐾)

**Assertion**
Ref	Expression
opltcon2b	⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ 𝑌 < ( ⊥ ‘𝑋)))

**Proof of Theorem opltcon2b**
Step	Hyp	Ref	Expression
1		opltcon3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		opltcon3.o	. . . . 5 ⊢ ⊥ = (oc‘𝐾)
3	1, 2	opoccl 39654	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
4	3	3adant2 1132	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
5		opltcon3.s	. . . 4 ⊢ < = (lt‘𝐾)
6	1, 5, 2	opltcon3b 39664	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) < ( ⊥ ‘𝑋)))
7	4, 6	syld3an3 1412	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) < ( ⊥ ‘𝑋)))
8	1, 2	opococ 39655	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
9	8	3adant2 1132	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
10	9	breq1d 5096	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘( ⊥ ‘𝑌)) < ( ⊥ ‘𝑋) ↔ 𝑌 < ( ⊥ ‘𝑋)))
11	7, 10	bitrd 279	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ 𝑌 < ( ⊥ ‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 class class class wbr 5086 ‘cfv 6492 Basecbs 17170 occoc 17219 ltcplt 18265 OPcops 39632

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pr 5370

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-iota 6448 df-fun 6494 df-fv 6500 df-ov 7363 df-proset 18251 df-poset 18270 df-plt 18285 df-oposet 39636

This theorem is referenced by: cvrcon3b 39737

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