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Mirrors > Home > MPE Home > Th. List > isposixOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of isposix 18219 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 5390 | . . 3 β’ {β¨(Baseβndx), π΅β©, β¨(leβndx), β€ β©} β V | |
3 | 1, 2 | eqeltri 2830 | . 2 β’ πΎ β V |
4 | isposix.a | . . 3 β’ π΅ β V | |
5 | df-ple 17158 | . . . 4 β’ le = Slot ;10 | |
6 | 1lt10 12762 | . . . 4 β’ 1 < ;10 | |
7 | 10nn 12639 | . . . 4 β’ ;10 β β | |
8 | 1, 5, 6, 7 | 2strbas 17111 | . . 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 17112 | . . 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 18218 | 1 β’ πΎ β Poset |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3444 {cpr 4589 β¨cop 4593 class class class wbr 5106 βcfv 6497 0cc0 11056 1c1 11057 ;cdc 12623 ndxcnx 17070 Basecbs 17088 lecple 17145 Posetcpo 18201 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-ple 17158 df-poset 18207 |
This theorem is referenced by: (None) |
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