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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 9187 | . 2 | |
2 | andi 808 | . . 3 | |
3 | df-3or 968 | . . . 4 | |
4 | 3 | anbi2i 453 | . . 3 |
5 | nn0re 9117 | . . . . . 6 | |
6 | 5 | pm4.71ri 390 | . . . . 5 |
7 | elnn0 9110 | . . . . . . 7 | |
8 | orcom 718 | . . . . . . 7 | |
9 | 7, 8 | bitri 183 | . . . . . 6 |
10 | 9 | anbi2i 453 | . . . . 5 |
11 | 6, 10 | bitri 183 | . . . 4 |
12 | 11 | orbi1i 753 | . . 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 698 w3o 966 wceq 1342 wcel 2135 cr 7746 cc0 7747 cneg 8064 cn 8851 cn0 9108 cz 9185 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4097 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-i2m1 7852 ax-rnegex 7856 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2726 df-un 3118 df-in 3120 df-ss 3127 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-int 3822 df-br 3980 df-iota 5150 df-fv 5193 df-ov 5842 df-neg 8066 df-inn 8852 df-n0 9109 df-z 9186 |
This theorem is referenced by: peano2z 9221 zindd 9303 expcl2lemap 10461 mulexpzap 10489 expaddzap 10493 expmulzap 10495 absexpzap 11016 pcid 12249 |
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