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Mirrors > Home > ILE Home > Th. List > peano2uz | Unicode version |
Description: Second Peano postulate for an upper set of integers. (Contributed by NM, 7-Sep-2005.) |
Ref | Expression |
---|---|
peano2uz |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 981 | . . 3 | |
2 | peano2z 9090 | . . . 4 | |
3 | 2 | 3ad2ant2 1003 | . . 3 |
4 | zre 9058 | . . . 4 | |
5 | zre 9058 | . . . . 5 | |
6 | letrp1 8606 | . . . . 5 | |
7 | 5, 6 | syl3an2 1250 | . . . 4 |
8 | 4, 7 | syl3an1 1249 | . . 3 |
9 | 1, 3, 8 | 3jca 1161 | . 2 |
10 | eluz2 9332 | . 2 | |
11 | eluz2 9332 | . 2 | |
12 | 9, 10, 11 | 3imtr4i 200 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 962 wcel 1480 class class class wbr 3929 cfv 5123 (class class class)co 5774 cr 7619 c1 7621 caddc 7623 cle 7801 cz 9054 cuz 9326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 |
This theorem is referenced by: peano2uzs 9379 peano2uzr 9380 uzaddcl 9381 fzsplit 9831 fzssp1 9847 fzsuc 9849 fzpred 9850 fzp1ss 9853 fzp1elp1 9855 fztp 9858 fzneuz 9881 fzosplitsnm1 9986 fzofzp1 10004 fzosplitsn 10010 fzostep1 10014 frec2uzuzd 10175 frecuzrdgrrn 10181 frec2uzrdg 10182 frecuzrdgrcl 10183 frecuzrdgsuc 10187 frecuzrdgrclt 10188 frecuzrdgg 10189 frecuzrdgsuctlem 10196 frecfzen2 10200 fzfig 10203 uzsinds 10215 iseqovex 10229 seq3val 10231 seqvalcd 10232 seqf 10234 seq3p1 10235 seq3split 10252 seq3homo 10283 seq3z 10284 ser3ge0 10290 faclbnd3 10489 bcm1k 10506 seq3coll 10585 clim2ser 11106 clim2ser2 11107 serf0 11121 fsump1 11189 fsump1i 11202 fsumparts 11239 isum1p 11261 cvgratnnlemmn 11294 mertenslemi1 11304 clim2prod 11308 clim2divap 11309 zsupcllemstep 11638 infssuzex 11642 |
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