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Mirrors > Home > MPE Home > Th. List > isposix | Structured version Visualization version GIF version |
Description: Properties that determine a poset (explicit structure version). Note that the numeric indices of the structure components are not mentioned explicitly in either the theorem or its proof. (Contributed by NM, 9-Nov-2012.) (Proof shortened by AV, 30-Oct-2024.) |
Ref | Expression |
---|---|
isposix.a | ⊢ 𝐵 ∈ V |
isposix.b | ⊢ ≤ ∈ V |
isposix.k | ⊢ 𝐾 = {〈(Base‘ndx), 𝐵〉, 〈(le‘ndx), ≤ 〉} |
isposix.1 | ⊢ (𝑥 ∈ 𝐵 → 𝑥 ≤ 𝑥) |
isposix.2 | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦)) |
isposix.3 | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) |
Ref | Expression |
---|---|
isposix | ⊢ 𝐾 ∈ Poset |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isposix.k | . . 3 ⊢ 𝐾 = {〈(Base‘ndx), 𝐵〉, 〈(le‘ndx), ≤ 〉} | |
2 | prex 5387 | . . 3 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(le‘ndx), ≤ 〉} ∈ V | |
3 | 1, 2 | eqeltri 2834 | . 2 ⊢ 𝐾 ∈ V |
4 | isposix.a | . . 3 ⊢ 𝐵 ∈ V | |
5 | basendxltplendx 17204 | . . . 4 ⊢ (Base‘ndx) < (le‘ndx) | |
6 | plendxnn 17203 | . . . 4 ⊢ (le‘ndx) ∈ ℕ | |
7 | 1, 5, 6 | 2strbas1 17064 | . . 3 ⊢ (𝐵 ∈ V → 𝐵 = (Base‘𝐾)) |
8 | 4, 7 | ax-mp 5 | . 2 ⊢ 𝐵 = (Base‘𝐾) |
9 | isposix.b | . . 3 ⊢ ≤ ∈ V | |
10 | pleid 17202 | . . . 4 ⊢ le = Slot (le‘ndx) | |
11 | 1, 5, 6, 10 | 2strop1 17065 | . . 3 ⊢ ( ≤ ∈ V → ≤ = (le‘𝐾)) |
12 | 9, 11 | ax-mp 5 | . 2 ⊢ ≤ = (le‘𝐾) |
13 | isposix.1 | . 2 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ≤ 𝑥) | |
14 | isposix.2 | . 2 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦)) | |
15 | isposix.3 | . 2 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) | |
16 | 3, 8, 12, 13, 14, 15 | isposi 18167 | 1 ⊢ 𝐾 ∈ Poset |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3443 {cpr 4586 〈cop 4590 class class class wbr 5103 ‘cfv 6493 ndxcnx 17019 Basecbs 17037 lecple 17094 Posetcpo 18150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-fz 13379 df-struct 16973 df-slot 17008 df-ndx 17020 df-base 17038 df-ple 17107 df-poset 18156 |
This theorem is referenced by: (None) |
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