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Mirrors > Home > ILE Home > Th. List > elz | Unicode version |
Description: Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) |
Ref | Expression |
---|---|
elz |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2177 | . . 3 | |
2 | eleq1 2233 | . . 3 | |
3 | negeq 8112 | . . . 4 | |
4 | 3 | eleq1d 2239 | . . 3 |
5 | 1, 2, 4 | 3orbi123d 1306 | . 2 |
6 | df-z 9213 | . 2 | |
7 | 5, 6 | elrab2 2889 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 w3o 972 wceq 1348 wcel 2141 cr 7773 cc0 7774 cneg 8091 cn 8878 cz 9212 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-fv 5206 df-ov 5856 df-neg 8093 df-z 9213 |
This theorem is referenced by: nnnegz 9215 zre 9216 elnnz 9222 0z 9223 elnn0z 9225 elznn0nn 9226 elznn0 9227 elznn 9228 znegcl 9243 zaddcl 9252 ztri3or0 9254 zeo 9317 addmodlteq 10354 zabsle1 13694 |
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