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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 | fvex 6113 | . . . 4 ⊢ (ODual‘𝑂) ∈ V | |
3 | 1, 2 | eqeltri 2684 | . . 3 ⊢ 𝐷 ∈ V |
4 | 3 | a1i 11 | . 2 ⊢ (𝑂 ∈ Poset → 𝐷 ∈ V) |
5 | eqid 2610 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
6 | 1, 5 | odubas 16956 | . . 3 ⊢ (Base‘𝑂) = (Base‘𝐷) |
7 | 6 | a1i 11 | . 2 ⊢ (𝑂 ∈ Poset → (Base‘𝑂) = (Base‘𝐷)) |
8 | eqid 2610 | . . . 4 ⊢ (le‘𝑂) = (le‘𝑂) | |
9 | 1, 8 | oduleval 16954 | . . 3 ⊢ ◡(le‘𝑂) = (le‘𝐷) |
10 | 9 | a1i 11 | . 2 ⊢ (𝑂 ∈ Poset → ◡(le‘𝑂) = (le‘𝐷)) |
11 | 5, 8 | posref 16774 | . . 3 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂)) → 𝑎(le‘𝑂)𝑎) |
12 | vex 3176 | . . . 4 ⊢ 𝑎 ∈ V | |
13 | 12, 12 | brcnv 5227 | . . 3 ⊢ (𝑎◡(le‘𝑂)𝑎 ↔ 𝑎(le‘𝑂)𝑎) |
14 | 11, 13 | sylibr 223 | . 2 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂)) → 𝑎◡(le‘𝑂)𝑎) |
15 | vex 3176 | . . . . 5 ⊢ 𝑏 ∈ V | |
16 | 12, 15 | brcnv 5227 | . . . 4 ⊢ (𝑎◡(le‘𝑂)𝑏 ↔ 𝑏(le‘𝑂)𝑎) |
17 | 15, 12 | brcnv 5227 | . . . 4 ⊢ (𝑏◡(le‘𝑂)𝑎 ↔ 𝑎(le‘𝑂)𝑏) |
18 | 16, 17 | anbi12ci 730 | . . 3 ⊢ ((𝑎◡(le‘𝑂)𝑏 ∧ 𝑏◡(le‘𝑂)𝑎) ↔ (𝑎(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎)) |
19 | 5, 8 | posasymb 16775 | . . . 4 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂)) → ((𝑎(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎) ↔ 𝑎 = 𝑏)) |
20 | 19 | biimpd 218 | . . 3 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂)) → ((𝑎(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎) → 𝑎 = 𝑏)) |
21 | 18, 20 | syl5bi 231 | . 2 ⊢ ((𝑂 ∈ Poset ∧ 𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂)) → ((𝑎◡(le‘𝑂)𝑏 ∧ 𝑏◡(le‘𝑂)𝑎) → 𝑎 = 𝑏)) |
22 | 3anrev 1042 | . . . 4 ⊢ ((𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂) ∧ 𝑐 ∈ (Base‘𝑂)) ↔ (𝑐 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂) ∧ 𝑎 ∈ (Base‘𝑂))) | |
23 | 5, 8 | postr 16776 | . . . 4 ⊢ ((𝑂 ∈ Poset ∧ (𝑐 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂) ∧ 𝑎 ∈ (Base‘𝑂))) → ((𝑐(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎) → 𝑐(le‘𝑂)𝑎)) |
24 | 22, 23 | sylan2b 491 | . . 3 ⊢ ((𝑂 ∈ Poset ∧ (𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂) ∧ 𝑐 ∈ (Base‘𝑂))) → ((𝑐(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎) → 𝑐(le‘𝑂)𝑎)) |
25 | vex 3176 | . . . . 5 ⊢ 𝑐 ∈ V | |
26 | 15, 25 | brcnv 5227 | . . . 4 ⊢ (𝑏◡(le‘𝑂)𝑐 ↔ 𝑐(le‘𝑂)𝑏) |
27 | 16, 26 | anbi12ci 730 | . . 3 ⊢ ((𝑎◡(le‘𝑂)𝑏 ∧ 𝑏◡(le‘𝑂)𝑐) ↔ (𝑐(le‘𝑂)𝑏 ∧ 𝑏(le‘𝑂)𝑎)) |
28 | 12, 25 | brcnv 5227 | . . 3 ⊢ (𝑎◡(le‘𝑂)𝑐 ↔ 𝑐(le‘𝑂)𝑎) |
29 | 24, 27, 28 | 3imtr4g 284 | . 2 ⊢ ((𝑂 ∈ Poset ∧ (𝑎 ∈ (Base‘𝑂) ∧ 𝑏 ∈ (Base‘𝑂) ∧ 𝑐 ∈ (Base‘𝑂))) → ((𝑎◡(le‘𝑂)𝑏 ∧ 𝑏◡(le‘𝑂)𝑐) → 𝑎◡(le‘𝑂)𝑐)) |
30 | 4, 7, 10, 14, 21, 29 | isposd 16778 | 1 ⊢ (𝑂 ∈ Poset → 𝐷 ∈ Poset) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 ◡ccnv 5037 ‘cfv 5804 Basecbs 15695 lecple 15775 Posetcpo 16763 ODualcodu 16951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-dec 11370 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ple 15788 df-preset 16751 df-poset 16769 df-odu 16952 |
This theorem is referenced by: oduposb 16959 posglbd 16973 odutos 28994 |
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