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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 28905 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 35262 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
4 | 3 | 3adant2 1165 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑌) ∈ 𝐵) |
5 | opltcon3.s | . . . 4 ⊢ < = (lt‘𝐾) | |
6 | 1, 5, 2 | opltcon3b 35272 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ ( ⊥ ‘𝑌) ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) < ( ⊥ ‘𝑋))) |
7 | 4, 6 | syld3an3 1532 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) < ( ⊥ ‘𝑋))) |
8 | 1, 2 | opococ 35263 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
9 | 8 | 3adant2 1165 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
10 | 9 | breq1d 4883 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘( ⊥ ‘𝑌)) < ( ⊥ ‘𝑋) ↔ 𝑌 < ( ⊥ ‘𝑋))) |
11 | 7, 10 | bitrd 271 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < ( ⊥ ‘𝑌) ↔ 𝑌 < ( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 class class class wbr 4873 ‘cfv 6123 Basecbs 16222 occoc 16313 ltcplt 17294 OPcops 35240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 df-ov 6908 df-proset 17281 df-poset 17299 df-plt 17311 df-oposet 35244 |
This theorem is referenced by: cvrcon3b 35345 |
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