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| Description: Being a poset is a self-dual property. (Contributed by Stefan O'Rear, 29-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| odupos.d | ⊢ 𝐷 = (ODual‘𝑂) | 
| Ref | Expression | 
|---|---|
| oduposb | ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | odupos.d | . . 3 ⊢ 𝐷 = (ODual‘𝑂) | |
| 2 | 1 | odupos 18373 | . 2 ⊢ (𝑂 ∈ Poset → 𝐷 ∈ Poset) | 
| 3 | eqid 2737 | . . . 4 ⊢ (ODual‘𝐷) = (ODual‘𝐷) | |
| 4 | 3 | odupos 18373 | . . 3 ⊢ (𝐷 ∈ Poset → (ODual‘𝐷) ∈ Poset) | 
| 5 | fvexd 6921 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (ODual‘𝐷) ∈ V) | |
| 6 | id 22 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝑉) | |
| 7 | eqid 2737 | . . . . . . 7 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
| 8 | 1, 7 | odubas 18336 | . . . . . 6 ⊢ (Base‘𝑂) = (Base‘𝐷) | 
| 9 | 3, 8 | odubas 18336 | . . . . 5 ⊢ (Base‘𝑂) = (Base‘(ODual‘𝐷)) | 
| 10 | 9 | a1i 11 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (Base‘𝑂) = (Base‘(ODual‘𝐷))) | 
| 11 | eqidd 2738 | . . . 4 ⊢ (𝑂 ∈ 𝑉 → (Base‘𝑂) = (Base‘𝑂)) | |
| 12 | eqid 2737 | . . . . . . . . . 10 ⊢ (le‘𝑂) = (le‘𝑂) | |
| 13 | 1, 12 | oduleval 18334 | . . . . . . . . 9 ⊢ ◡(le‘𝑂) = (le‘𝐷) | 
| 14 | 3, 13 | oduleval 18334 | . . . . . . . 8 ⊢ ◡◡(le‘𝑂) = (le‘(ODual‘𝐷)) | 
| 15 | 14 | eqcomi 2746 | . . . . . . 7 ⊢ (le‘(ODual‘𝐷)) = ◡◡(le‘𝑂) | 
| 16 | 15 | breqi 5149 | . . . . . 6 ⊢ (𝑎(le‘(ODual‘𝐷))𝑏 ↔ 𝑎◡◡(le‘𝑂)𝑏) | 
| 17 | vex 3484 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
| 18 | vex 3484 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
| 19 | 17, 18 | brcnv 5893 | . . . . . 6 ⊢ (𝑎◡◡(le‘𝑂)𝑏 ↔ 𝑏◡(le‘𝑂)𝑎) | 
| 20 | 18, 17 | brcnv 5893 | . . . . . 6 ⊢ (𝑏◡(le‘𝑂)𝑎 ↔ 𝑎(le‘𝑂)𝑏) | 
| 21 | 16, 19, 20 | 3bitri 297 | . . . . 5 ⊢ (𝑎(le‘(ODual‘𝐷))𝑏 ↔ 𝑎(le‘𝑂)𝑏) | 
| 22 | 21 | a1i 11 | . . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ (𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂))) → (𝑎(le‘(ODual‘𝐷))𝑏 ↔ 𝑎(le‘𝑂)𝑏)) | 
| 23 | 5, 6, 10, 11, 22 | pospropd 18372 | . . 3 ⊢ (𝑂 ∈ 𝑉 → ((ODual‘𝐷) ∈ Poset ↔ 𝑂 ∈ Poset)) | 
| 24 | 4, 23 | imbitrid 244 | . 2 ⊢ (𝑂 ∈ 𝑉 → (𝐷 ∈ Poset → 𝑂 ∈ Poset)) | 
| 25 | 2, 24 | impbid2 226 | 1 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ Poset ↔ 𝐷 ∈ Poset)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 ◡ccnv 5684 ‘cfv 6561 Basecbs 17247 lecple 17304 ODualcodu 18331 Posetcpo 18353 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-dec 12734 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ple 17317 df-odu 18332 df-proset 18340 df-poset 18359 | 
| This theorem is referenced by: odulatb 18479 oduclatb 18552 | 
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