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Mirrors > Home > MPE Home > Th. List > Mathboxes > oplecon1b | Structured version Visualization version GIF version |
Description: Contraposition law for strict ordering in orthoposets. (chsscon1 31383 analog.) (Contributed by NM, 6-Nov-2011.) |
Ref | Expression |
---|---|
opcon3.b | ⊢ 𝐵 = (Base‘𝐾) |
opcon3.l | ⊢ ≤ = (le‘𝐾) |
opcon3.o | ⊢ ⊥ = (oc‘𝐾) |
Ref | Expression |
---|---|
oplecon1b | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opcon3.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | opcon3.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
3 | 1, 2 | opoccl 38793 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
4 | 3 | 3adant3 1129 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
5 | opcon3.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
6 | 1, 5, 2 | oplecon3b 38799 | . . 3 ⊢ ((𝐾 ∈ OP ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘( ⊥ ‘𝑋)))) |
7 | 4, 6 | syld3an2 1408 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘( ⊥ ‘𝑋)))) |
8 | 1, 2 | opococ 38794 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
9 | 8 | 3adant3 1129 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
10 | 9 | breq2d 5161 | . 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 5149 ‘cfv 6549 Basecbs 17183 lecple 17243 occoc 17244 OPcops 38771 |
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-ext 2696 ax-nul 5307 |
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-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-dm 5688 df-iota 6501 df-fv 6557 df-ov 7422 df-oposet 38775 |
This theorem is referenced by: opoc1 38801 oldmm1 38816 |
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