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Mirrors > Home > ILE Home > Th. List > nnuz | Unicode version |
Description: Positive integers expressed as an upper set of integers. (Contributed by NM, 2-Sep-2005.) |
Ref | Expression |
---|---|
nnuz |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnzrab 9290 |
. 2
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2 | 1z 9292 |
. . 3
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3 | uzval 9543 |
. . 3
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4 | 2, 3 | ax-mp 5 |
. 2
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5 | 1, 4 | eqtr4i 2211 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-inn 8933 df-z 9267 df-uz 9542 |
This theorem is referenced by: elnnuz 9577 eluz2nn 9579 uznnssnn 9590 eluznn 9613 fzssnn 10081 fseq1p1m1 10107 fz01or 10124 nnsplit 10150 elfzo1 10203 exp3vallem 10534 exp3val 10535 facnn 10720 fac0 10721 bcm1k 10753 bcval5 10756 bcpasc 10759 seq3coll 10835 recvguniq 11017 resqrexlemf 11029 climuni 11314 climrecvg1n 11369 climcvg1nlem 11370 summodclem3 11401 summodclem2a 11402 fsum3 11408 sum0 11409 isumz 11410 fsumcl2lem 11419 fsumadd 11427 fsummulc2 11469 isumnn0nn 11514 divcnv 11518 trireciplem 11521 trirecip 11522 expcnvap0 11523 expcnv 11525 geo2lim 11537 geoisum1 11540 geoisum1c 11541 cvgratnnlemnexp 11545 cvgratnnlemseq 11547 cvgratnnlemrate 11551 cvgratnn 11552 mertenslem2 11557 prodmodclem3 11596 prodmodclem2a 11597 fprodseq 11604 prod0 11606 prod1dc 11607 fprodssdc 11611 fprodmul 11612 ege2le3 11692 nninfdcex 11967 gcdsupex 11971 gcdsupcl 11972 nnmindc 12048 nnminle 12049 lcmval 12076 lcmcllem 12080 lcmledvds 12083 isprm3 12131 phicl2 12227 phibndlem 12229 odzcllem 12255 odzdvds 12258 pcmptcl 12353 pcmpt 12354 pockthlem 12367 pockthg 12368 1arith 12378 ennnfonelemjn 12416 ssnnctlemct 12460 nninfdclemf 12463 nninfdclemp1 12464 mulgval 13016 mulgfng 13018 mulgnnp1 13022 mulgnnsubcl 13026 mulgnn0z 13041 mulgnndir 13043 mulgpropdg 13056 lmtopcnp 14021 lgsval 14676 lgscllem 14679 lgsval2lem 14682 lgsval4a 14694 lgsneg 14696 lgsdir 14707 lgsdilem2 14708 lgsdi 14709 lgsne0 14710 cvgcmp2nlemabs 15052 cvgcmp2n 15053 trilpolemcl 15057 trilpolemisumle 15058 trilpolemgt1 15059 trilpolemeq1 15060 trilpolemlt1 15061 nconstwlpolem0 15083 nconstwlpolemgt0 15084 |
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