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Mirrors > Home > MPE Home > Th. List > isposixOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of isposix 18290 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 5425 | . . 3 β’ {β¨(Baseβndx), π΅β©, β¨(leβndx), β€ β©} β V | |
3 | 1, 2 | eqeltri 2823 | . 2 β’ πΎ β V |
4 | isposix.a | . . 3 β’ π΅ β V | |
5 | df-ple 17226 | . . . 4 β’ le = Slot ;10 | |
6 | 1lt10 12820 | . . . 4 β’ 1 < ;10 | |
7 | 10nn 12697 | . . . 4 β’ ;10 β β | |
8 | 1, 5, 6, 7 | 2strbas 17176 | . . 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 17177 | . . 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 18289 | 1 β’ πΎ β Poset |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3468 {cpr 4625 β¨cop 4629 class class class wbr 5141 βcfv 6537 0cc0 11112 1c1 11113 ;cdc 12681 ndxcnx 17135 Basecbs 17153 lecple 17213 Posetcpo 18272 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-ple 17226 df-poset 18278 |
This theorem is referenced by: (None) |
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