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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 28916 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 35270 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
4 | 3 | 3adant3 1168 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘𝑋) ∈ 𝐵) |
5 | opcon3.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
6 | 1, 5, 2 | oplecon3b 35276 | . . 3 ⊢ ((𝐾 ∈ OP ∧ ( ⊥ ‘𝑋) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘( ⊥ ‘𝑋)))) |
7 | 4, 6 | syld3an2 1537 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (( ⊥ ‘𝑋) ≤ 𝑌 ↔ ( ⊥ ‘𝑌) ≤ ( ⊥ ‘( ⊥ ‘𝑋)))) |
8 | 1, 2 | opococ 35271 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
9 | 8 | 3adant3 1168 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
10 | 9 | breq2d 4886 | . 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 1113 = wceq 1658 ∈ wcel 2166 class class class wbr 4874 ‘cfv 6124 Basecbs 16223 lecple 16313 occoc 16314 OPcops 35248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-nul 5014 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-dm 5353 df-iota 6087 df-fv 6132 df-ov 6909 df-oposet 35252 |
This theorem is referenced by: opoc1 35278 oldmm1 35293 |
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