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Mirrors > Home > ILE Home > Th. List > elznn0nn | Unicode version |
Description: Integer property expressed in terms nonnegative integers and positive integers. (Contributed by NM, 10-May-2004.) |
Ref | Expression |
---|---|
elznn0nn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 9024 | . 2 | |
2 | andi 792 | . . 3 | |
3 | df-3or 948 | . . . 4 | |
4 | 3 | anbi2i 452 | . . 3 |
5 | nn0re 8954 | . . . . . 6 | |
6 | 5 | pm4.71ri 389 | . . . . 5 |
7 | elnn0 8947 | . . . . . . 7 | |
8 | orcom 702 | . . . . . . 7 | |
9 | 7, 8 | bitri 183 | . . . . . 6 |
10 | 9 | anbi2i 452 | . . . . 5 |
11 | 6, 10 | bitri 183 | . . . 4 |
12 | 11 | orbi1i 737 | . . 3 |
13 | 2, 4, 12 | 3bitr4i 211 | . 2 |
14 | 1, 13 | bitri 183 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wo 682 w3o 946 wceq 1316 wcel 1465 cr 7587 cc0 7588 cneg 7902 cn 8688 cn0 8945 cz 9022 |
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-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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 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-i2m1 7693 ax-rnegex 7697 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 |
This theorem is referenced by: peano2z 9058 zindd 9137 expcl2lemap 10273 mulexpzap 10301 expaddzap 10305 expmulzap 10307 absexpzap 10820 |
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