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Mirrors > Home > MPE Home > Th. List > odupos | Structured version Visualization version GIF version |
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 |
---|---|
odupos | ⊢ (𝑂 ∈ Poset → 𝐷 ∈ Poset) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | odupos.d | . . . 4 ⊢ 𝐷 = (ODual‘𝑂) | |
2 | 1 | fvexi 6447 | . . 3 ⊢ 𝐷 ∈ V |
3 | 2 | a1i 11 | . 2 ⊢ (𝑂 ∈ Poset → 𝐷 ∈ V) |
4 | eqid 2825 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
5 | 1, 4 | odubas 17486 | . . 3 ⊢ (Base‘𝑂) = (Base‘𝐷) |
6 | 5 | a1i 11 | . 2 ⊢ (𝑂 ∈ Poset → (Base‘𝑂) = (Base‘𝐷)) |
7 | eqid 2825 | . . . 4 ⊢ (le‘𝑂) = (le‘𝑂) | |
8 | 1, 7 | oduleval 17484 | . . 3 ⊢ ◡(le‘𝑂) = (le‘𝐷) |
9 | 8 | a1i 11 | . 2 ⊢ (𝑂 ∈ Poset → ◡(le‘𝑂) = (le‘𝐷)) |
10 | 4, 7 | posref 17304 | . . 3 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂)) → 𝑎(le‘𝑂)𝑎) |
11 | vex 3417 | . . . 4 ⊢ 𝑎 ∈ V | |
12 | 11, 11 | brcnv 5537 | . . 3 ⊢ (𝑎◡(le‘𝑂)𝑎 ↔ 𝑎(le‘𝑂)𝑎) |
13 | 10, 12 | sylibr 226 | . 2 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂)) → 𝑎◡(le‘𝑂)𝑎) |
14 | vex 3417 | . . . . 5 ⊢ 𝑏 ∈ V | |
15 | 11, 14 | brcnv 5537 | . . . 4 ⊢ (𝑎◡(le‘𝑂)𝑏 ↔ 𝑏(le‘𝑂)𝑎) |
16 | 14, 11 | brcnv 5537 | . . . 4 ⊢ (𝑏◡(le‘𝑂)𝑎 ↔ 𝑎(le‘𝑂)𝑏) |
17 | 15, 16 | anbi12ci 623 | . . 3 ⊢ ((𝑎◡(le‘𝑂)𝑏 ∧ 𝑏◡(le‘𝑂)𝑎) ↔ (𝑎(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎)) |
18 | 4, 7 | posasymb 17305 | . . . 4 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂)) → ((𝑎(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎) ↔ 𝑎 = 𝑏)) |
19 | 18 | biimpd 221 | . . 3 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂)) → ((𝑎(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎) → 𝑎 = 𝑏)) |
20 | 17, 19 | syl5bi 234 | . 2 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂)) → ((𝑎◡(le‘𝑂)𝑏 ∧ 𝑏◡(le‘𝑂)𝑎) → 𝑎 = 𝑏)) |
21 | 3anrev 1132 | . . . 4 ⊢ ((𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂) ∧ 𝑐 ∈ (Base‘𝑂)) ↔ (𝑐 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂) ∧ 𝑎 ∈ (Base‘𝑂))) | |
22 | 4, 7 | postr 17306 | . . . 4 ⊢ ((𝑂 ∈ Poset ∧ (𝑐 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂) ∧ 𝑎 ∈ (Base‘𝑂))) → ((𝑐(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎) → 𝑐(le‘𝑂)𝑎)) |
23 | 21, 22 | sylan2b 589 | . . 3 ⊢ ((𝑂 ∈ Poset ∧ (𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂) ∧ 𝑐 ∈ (Base‘𝑂))) → ((𝑐(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎) → 𝑐(le‘𝑂)𝑎)) |
24 | vex 3417 | . . . . 5 ⊢ 𝑐 ∈ V | |
25 | 14, 24 | brcnv 5537 | . . . 4 ⊢ (𝑏◡(le‘𝑂)𝑐 ↔ 𝑐(le‘𝑂)𝑏) |
26 | 15, 25 | anbi12ci 623 | . . 3 ⊢ ((𝑎◡(le‘𝑂)𝑏 ∧ 𝑏◡(le‘𝑂)𝑐) ↔ (𝑐(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎)) |
27 | 11, 24 | brcnv 5537 | . . 3 ⊢ (𝑎◡(le‘𝑂)𝑐 ↔ 𝑐(le‘𝑂)𝑎) |
28 | 23, 26, 27 | 3imtr4g 288 | . 2 ⊢ ((𝑂 ∈ Poset ∧ (𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂) ∧ 𝑐 ∈ (Base‘𝑂))) → ((𝑎◡(le‘𝑂)𝑏 ∧ 𝑏◡(le‘𝑂)𝑐) → 𝑎◡(le‘𝑂)𝑐)) |
29 | 3, 6, 9, 13, 20, 28 | isposd 17308 | 1 ⊢ (𝑂 ∈ Poset → 𝐷 ∈ Poset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 Vcvv 3414 class class class wbr 4873 ◡ccnv 5341 ‘cfv 6123 Basecbs 16222 lecple 16312 Posetcpo 17293 ODualcodu 17481 |
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-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-dec 11822 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ple 16325 df-proset 17281 df-poset 17299 df-odu 17482 |
This theorem is referenced by: oduposb 17489 posglbd 17503 odutos 30208 |
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