![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isposixOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of isposix 18273 as of 30-Oct-2024. 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 (Remark: That is not true - it becomes true with the new proof!). (Contributed by NM, 9-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
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 |
---|---|
isposixOLD | ⊢ 𝐾 ∈ Poset |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isposix.k | . . 3 ⊢ 𝐾 = {〈(Base‘ndx), 𝐵〉, 〈(le‘ndx), ≤ 〉} | |
2 | prex 5430 | . . 3 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(le‘ndx), ≤ 〉} ∈ V | |
3 | 1, 2 | eqeltri 2830 | . 2 ⊢ 𝐾 ∈ V |
4 | isposix.a | . . 3 ⊢ 𝐵 ∈ V | |
5 | df-ple 17212 | . . . 4 ⊢ le = Slot ;10 | |
6 | 1lt10 12811 | . . . 4 ⊢ 1 < ;10 | |
7 | 10nn 12688 | . . . 4 ⊢ ;10 ∈ ℕ | |
8 | 1, 5, 6, 7 | 2strbas 17162 | . . 3 ⊢ (𝐵 ∈ V → 𝐵 = (Base‘𝐾)) |
9 | 4, 8 | ax-mp 5 | . 2 ⊢ 𝐵 = (Base‘𝐾) |
10 | isposix.b | . . 3 ⊢ ≤ ∈ V | |
11 | 1, 5, 6, 7 | 2strop 17163 | . . 3 ⊢ ( ≤ ∈ V → ≤ = (le‘𝐾)) |
12 | 10, 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, 9, 12, 13, 14, 15 | isposi 18272 | 1 ⊢ 𝐾 ∈ Poset |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3475 {cpr 4628 〈cop 4632 class class class wbr 5146 ‘cfv 6539 0cc0 11105 1c1 11106 ;cdc 12672 ndxcnx 17121 Basecbs 17139 lecple 17199 Posetcpo 18255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12468 df-z 12554 df-dec 12673 df-uz 12818 df-fz 13480 df-struct 17075 df-slot 17110 df-ndx 17122 df-base 17140 df-ple 17212 df-poset 18261 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |