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Mirrors > Home > ILE Home > Th. List > eluzp1p1 | Unicode version |
Description: Membership in the next upper set of integers. (Contributed by NM, 5-Oct-2005.) |
Ref | Expression |
---|---|
eluzp1p1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2z 9058 | . . . 4 | |
2 | 1 | 3ad2ant1 987 | . . 3 |
3 | peano2z 9058 | . . . 4 | |
4 | 3 | 3ad2ant2 988 | . . 3 |
5 | zre 9026 | . . . . 5 | |
6 | zre 9026 | . . . . 5 | |
7 | 1re 7733 | . . . . . 6 | |
8 | leadd1 8160 | . . . . . 6 | |
9 | 7, 8 | mp3an3 1289 | . . . . 5 |
10 | 5, 6, 9 | syl2an 287 | . . . 4 |
11 | 10 | biimp3a 1308 | . . 3 |
12 | 2, 4, 11 | 3jca 1146 | . 2 |
13 | eluz2 9300 | . 2 | |
14 | eluz2 9300 | . 2 | |
15 | 12, 13, 14 | 3imtr4i 200 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 947 wcel 1465 class class class wbr 3899 cfv 5093 (class class class)co 5742 cr 7587 c1 7589 caddc 7591 cle 7769 cz 9022 cuz 9294 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 df-uz 9295 |
This theorem is referenced by: uzp1 9327 fzp1elp1 9823 rebtwn2z 10000 seqvalcd 10200 seqovcd 10204 seqp1cd 10207 seq3fveq2 10210 seq3id2 10250 seq3coll 10553 serf0 11089 efcllemp 11291 prmind2 11728 cvgcmp2nlemabs 13154 |
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