opltcon2b - Mathbox for Norm Megill

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Theorem opltcon2b 36341

Description: Contraposition law for strict ordering in orthoposets. (chsscon2 29278 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 36329	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
4	3	3adant2 1127	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
5		opltcon3.s	. . . 4 ⊢ < = (lt‘𝐾)
6	1, 5, 2	opltcon3b 36339	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) < ( ⊥ ‘𝑋)))
7	4, 6	syld3an3 1405	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) < ( ⊥ ‘𝑋)))
8	1, 2	opococ 36330	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
9	8	3adant2 1127	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
10	9	breq1d 5075	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘( ⊥ ‘𝑌)) < ( ⊥ ‘𝑋) ↔ 𝑌 < ( ⊥ ‘𝑋)))
11	7, 10	bitrd 281	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ 𝑌 < ( ⊥ ‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 ‘cfv 6354 Basecbs 16482 occoc 16572 ltcplt 17550 OPcops 36307

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-proset 17537 df-poset 17555 df-plt 17567 df-oposet 36311

This theorem is referenced by: cvrcon3b 36412

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