opltcon2b - Mathbox for Norm Megill

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Theorem opltcon2b 38734

Description: Contraposition law for strict ordering in orthoposets. (chsscon2 31356 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 38722	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
4	3	3adant2 1128	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵)
5		opltcon3.s	. . . 4 ⊢ < = (lt‘𝐾)
6	1, 5, 2	opltcon3b 38732	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) < ( ⊥ ‘𝑋)))
7	4, 6	syld3an3 1406	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) < ( ⊥ ‘𝑋)))
8	1, 2	opococ 38723	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
9	8	3adant2 1128	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)
10	9	breq1d 5153	. 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘( ⊥ ‘𝑌)) < ( ⊥ ‘𝑋) ↔ 𝑌 < ( ⊥ ‘𝑋)))
11	7, 10	bitrd 278	1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ 𝑌 < ( ⊥ ‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5143 ‘cfv 6543 Basecbs 17179 occoc 17240 ltcplt 18299 OPcops 38700

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7419 df-proset 18286 df-poset 18304 df-plt 18321 df-oposet 38704

This theorem is referenced by: cvrcon3b 38805

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