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Mirrors > Home > ILE Home > Th. List > eluzp1m1 | Unicode version |
Description: Membership in the next upper set of integers. (Contributed by NM, 12-Sep-2005.) |
Ref | Expression |
---|---|
eluzp1m1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2zm 9116 | . . . . . 6 | |
2 | 1 | ad2antrl 482 | . . . . 5 |
3 | zre 9082 | . . . . . . . 8 | |
4 | zre 9082 | . . . . . . . 8 | |
5 | 1re 7789 | . . . . . . . . 9 | |
6 | leaddsub 8224 | . . . . . . . . 9 | |
7 | 5, 6 | mp3an2 1304 | . . . . . . . 8 |
8 | 3, 4, 7 | syl2an 287 | . . . . . . 7 |
9 | 8 | biimpa 294 | . . . . . 6 |
10 | 9 | anasss 397 | . . . . 5 |
11 | 2, 10 | jca 304 | . . . 4 |
12 | 11 | ex 114 | . . 3 |
13 | peano2z 9114 | . . . 4 | |
14 | eluz1 9354 | . . . 4 | |
15 | 13, 14 | syl 14 | . . 3 |
16 | eluz1 9354 | . . 3 | |
17 | 12, 15, 16 | 3imtr4d 202 | . 2 |
18 | 17 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1481 class class class wbr 3937 cfv 5131 (class class class)co 5782 cr 7643 c1 7645 caddc 7647 cle 7825 cmin 7957 cz 9078 cuz 9350 |
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-iota 5096 df-fun 5133 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: peano2uzr 9407 fzosplitsnm1 10017 fzofzp1b 10036 seq3m1 10272 monoord 10280 seq3id 10312 seq3z 10315 serf0 11153 fsumm1 11217 telfsumo 11267 fsumparts 11271 isumsplit 11292 ennnfonelemjn 11951 |
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