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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 8930 |
. 2
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2 | 1z 8932 |
. . 3
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3 | uzval 9178 |
. . 3
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4 | 2, 3 | ax-mp 7 |
. 2
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5 | 1, 4 | eqtr4i 2123 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-addcom 7595 ax-addass 7597 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-0id 7603 ax-rnegex 7604 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-ltadd 7611 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-inn 8579 df-z 8907 df-uz 9177 |
This theorem is referenced by: elnnuz 9212 eluz2nn 9214 uznnssnn 9222 eluznn 9244 fzssnn 9689 fseq1p1m1 9715 fz01or 9732 nnsplit 9755 elfzo1 9808 exp3vallem 10135 exp3val 10136 facnn 10314 fac0 10315 bcm1k 10347 bcval5 10350 bcpasc 10353 seq3coll 10426 recvguniq 10607 resqrexlemf 10619 climuni 10901 climrecvg1n 10956 climcvg1nlem 10957 summodclem3 10988 summodclem2a 10989 fsum3 10995 sum0 10996 isumz 10997 fsumcl2lem 11006 fsumadd 11014 fsummulc2 11056 isumnn0nn 11101 divcnv 11105 trireciplem 11108 trirecip 11109 expcnvap0 11110 expcnv 11112 geo2lim 11124 geoisum1 11127 geoisum1c 11128 cvgratnnlemnexp 11132 cvgratnnlemseq 11134 cvgratnnlemrate 11138 cvgratnn 11139 mertenslem2 11144 ege2le3 11175 gcdsupex 11441 gcdsupcl 11442 lcmval 11537 lcmcllem 11541 lcmledvds 11544 isprm3 11592 phicl2 11682 phibndlem 11684 ennnfonelemjn 11707 lmtopcnp 12200 cvgcmp2nlemabs 12811 cvgcmp2n 12812 trilpolemcl 12814 trilpolemisumle 12815 trilpolemgt1 12816 trilpolemeq1 12817 trilpolemlt1 12818 |
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