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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 982 |
. . 3
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2 | peano2z 9114 |
. . . 4
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3 | 2 | 3ad2ant2 1004 |
. . 3
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4 | zre 9082 |
. . . 4
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5 | zre 9082 |
. . . . 5
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6 | letrp1 8630 |
. . . . 5
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7 | 5, 6 | syl3an2 1251 |
. . . 4
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8 | 4, 7 | syl3an1 1250 |
. . 3
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9 | 1, 3, 8 | 3jca 1162 |
. 2
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10 | eluz2 9356 |
. 2
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11 | eluz2 9356 |
. 2
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12 | 9, 10, 11 | 3imtr4i 200 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 |
This theorem is referenced by: peano2uzs 9406 peano2uzr 9407 uzaddcl 9408 fzsplit 9862 fzssp1 9878 fzsuc 9880 fzpred 9881 fzp1ss 9884 fzp1elp1 9886 fztp 9889 fzneuz 9912 fzosplitsnm1 10017 fzofzp1 10035 fzosplitsn 10041 fzostep1 10045 frec2uzuzd 10206 frecuzrdgrrn 10212 frec2uzrdg 10213 frecuzrdgrcl 10214 frecuzrdgsuc 10218 frecuzrdgrclt 10219 frecuzrdgg 10220 frecuzrdgsuctlem 10227 frecfzen2 10231 fzfig 10234 uzsinds 10246 iseqovex 10260 seq3val 10262 seqvalcd 10263 seqf 10265 seq3p1 10266 seq3split 10283 seq3homo 10314 seq3z 10315 ser3ge0 10321 faclbnd3 10521 bcm1k 10538 seq3coll 10617 clim2ser 11138 clim2ser2 11139 serf0 11153 fsump1 11221 fsump1i 11234 fsumparts 11271 isum1p 11293 cvgratnnlemmn 11326 mertenslemi1 11336 clim2prod 11340 clim2divap 11341 zsupcllemstep 11674 infssuzex 11678 |
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