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Mirrors > Home > MPE Home > Th. List > isposixOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of isposix 18316 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 | β’ π΅ β V |
isposix.b | β’ β€ β V |
isposix.k | β’ πΎ = {β¨(Baseβndx), π΅β©, β¨(leβndx), β€ β©} |
isposix.1 | β’ (π₯ β π΅ β π₯ β€ π₯) |
isposix.2 | β’ ((π₯ β π΅ β§ π¦ β π΅) β ((π₯ β€ π¦ β§ π¦ β€ π₯) β π₯ = π¦)) |
isposix.3 | β’ ((π₯ β π΅ β§ π¦ β π΅ β§ π§ β π΅) β ((π₯ β€ π¦ β§ π¦ β€ π§) β π₯ β€ π§)) |
Ref | Expression |
---|---|
isposixOLD | β’ πΎ β Poset |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isposix.k | . . 3 β’ πΎ = {β¨(Baseβndx), π΅β©, β¨(leβndx), β€ β©} | |
2 | prex 5428 | . . 3 β’ {β¨(Baseβndx), π΅β©, β¨(leβndx), β€ β©} β V | |
3 | 1, 2 | eqeltri 2821 | . 2 β’ πΎ β V |
4 | isposix.a | . . 3 β’ π΅ β V | |
5 | df-ple 17252 | . . . 4 β’ le = Slot ;10 | |
6 | 1lt10 12846 | . . . 4 β’ 1 < ;10 | |
7 | 10nn 12723 | . . . 4 β’ ;10 β β | |
8 | 1, 5, 6, 7 | 2strbas 17202 | . . 3 β’ (π΅ β V β π΅ = (BaseβπΎ)) |
9 | 4, 8 | ax-mp 5 | . 2 β’ π΅ = (BaseβπΎ) |
10 | isposix.b | . . 3 β’ β€ β V | |
11 | 1, 5, 6, 7 | 2strop 17203 | . . 3 β’ ( β€ β V β β€ = (leβπΎ)) |
12 | 10, 11 | ax-mp 5 | . 2 β’ β€ = (leβπΎ) |
13 | isposix.1 | . 2 β’ (π₯ β π΅ β π₯ β€ π₯) | |
14 | isposix.2 | . 2 β’ ((π₯ β π΅ β§ π¦ β π΅) β ((π₯ β€ π¦ β§ π¦ β€ π₯) β π₯ = π¦)) | |
15 | isposix.3 | . 2 β’ ((π₯ β π΅ β§ π¦ β π΅ β§ π§ β π΅) β ((π₯ β€ π¦ β§ π¦ β€ π§) β π₯ β€ π§)) | |
16 | 3, 9, 12, 13, 14, 15 | isposi 18315 | 1 β’ πΎ β Poset |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3463 {cpr 4626 β¨cop 4630 class class class wbr 5143 βcfv 6543 0cc0 11138 1c1 11139 ;cdc 12707 ndxcnx 17161 Basecbs 17179 lecple 17239 Posetcpo 18298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17180 df-ple 17252 df-poset 18304 |
This theorem is referenced by: (None) |
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