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Mirrors > Home > MPE Home > Th. List > Mathboxes > opltcon2b | Structured version Visualization version GIF version |
Description: Contraposition law for strict ordering in orthoposets. (chsscon2 29273 analog.) (Contributed by NM, 5-Nov-2011.) |
Ref | Expression |
---|---|
opltcon3.b | ⊢ 𝐵 = (Base‘𝐾) |
opltcon3.s | ⊢ < = (lt‘𝐾) |
opltcon3.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
opltcon2b | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ 𝑌 < ( ⊥ ‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opltcon3.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | opltcon3.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
3 | 1, 2 | opoccl 36324 | . . . 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 36334 | . . 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 36325 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
9 | 8 | 3adant2 1127 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
10 | 9 | breq1d 5068 | . 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 5058 ‘cfv 6349 Basecbs 16477 occoc 16567 ltcplt 17545 OPcops 36302 |
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 5195 ax-nul 5202 ax-pr 5321 |
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 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-ov 7153 df-proset 17532 df-poset 17550 df-plt 17562 df-oposet 36306 |
This theorem is referenced by: cvrcon3b 36407 |
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