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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 6677 | . . 3 ⊢ 𝐷 ∈ V |
3 | 2 | a1i 11 | . 2 ⊢ (𝑂 ∈ Poset → 𝐷 ∈ V) |
4 | eqid 2818 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
5 | 1, 4 | odubas 17731 | . . 3 ⊢ (Base‘𝑂) = (Base‘𝐷) |
6 | 5 | a1i 11 | . 2 ⊢ (𝑂 ∈ Poset → (Base‘𝑂) = (Base‘𝐷)) |
7 | eqid 2818 | . . . 4 ⊢ (le‘𝑂) = (le‘𝑂) | |
8 | 1, 7 | oduleval 17729 | . . 3 ⊢ ◡(le‘𝑂) = (le‘𝐷) |
9 | 8 | a1i 11 | . 2 ⊢ (𝑂 ∈ Poset → ◡(le‘𝑂) = (le‘𝐷)) |
10 | 4, 7 | posref 17549 | . . 3 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂)) → 𝑎(le‘𝑂)𝑎) |
11 | vex 3495 | . . . 4 ⊢ 𝑎 ∈ V | |
12 | 11, 11 | brcnv 5746 | . . 3 ⊢ (𝑎◡(le‘𝑂)𝑎 ↔ 𝑎(le‘𝑂)𝑎) |
13 | 10, 12 | sylibr 235 | . 2 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂)) → 𝑎◡(le‘𝑂)𝑎) |
14 | vex 3495 | . . . . 5 ⊢ 𝑏 ∈ V | |
15 | 11, 14 | brcnv 5746 | . . . 4 ⊢ (𝑎◡(le‘𝑂)𝑏 ↔ 𝑏(le‘𝑂)𝑎) |
16 | 14, 11 | brcnv 5746 | . . . 4 ⊢ (𝑏◡(le‘𝑂)𝑎 ↔ 𝑎(le‘𝑂)𝑏) |
17 | 15, 16 | anbi12ci 627 | . . 3 ⊢ ((𝑎◡(le‘𝑂)𝑏 ∧ 𝑏◡(le‘𝑂)𝑎) ↔ (𝑎(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎)) |
18 | 4, 7 | posasymb 17550 | . . . 4 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂)) → ((𝑎(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎) ↔ 𝑎 = 𝑏)) |
19 | 18 | biimpd 230 | . . 3 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂)) → ((𝑎(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎) → 𝑎 = 𝑏)) |
20 | 17, 19 | syl5bi 243 | . 2 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂)) → ((𝑎◡(le‘𝑂)𝑏 ∧ 𝑏◡(le‘𝑂)𝑎) → 𝑎 = 𝑏)) |
21 | 3anrev 1093 | . . . 4 ⊢ ((𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂) ∧ 𝑐 ∈ (Base‘𝑂)) ↔ (𝑐 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂) ∧ 𝑎 ∈ (Base‘𝑂))) | |
22 | 4, 7 | postr 17551 | . . . 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 593 | . . 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 3495 | . . . . 5 ⊢ 𝑐 ∈ V | |
25 | 14, 24 | brcnv 5746 | . . . 4 ⊢ (𝑏◡(le‘𝑂)𝑐 ↔ 𝑐(le‘𝑂)𝑏) |
26 | 15, 25 | anbi12ci 627 | . . 3 ⊢ ((𝑎◡(le‘𝑂)𝑏 ∧ 𝑏◡(le‘𝑂)𝑐) ↔ (𝑐(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎)) |
27 | 11, 24 | brcnv 5746 | . . 3 ⊢ (𝑎◡(le‘𝑂)𝑐 ↔ 𝑐(le‘𝑂)𝑎) |
28 | 23, 26, 27 | 3imtr4g 297 | . 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 17553 | 1 ⊢ (𝑂 ∈ Poset → 𝐷 ∈ Poset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3492 class class class wbr 5057 ◡ccnv 5547 ‘cfv 6348 Basecbs 16471 lecple 16560 Posetcpo 17538 ODualcodu 17726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-dec 12087 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ple 16573 df-proset 17526 df-poset 17544 df-odu 17727 |
This theorem is referenced by: oduposb 17734 posglbd 17748 odutos 30577 |
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