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Mirrors > Home > MPE Home > Th. List > elz | Structured version Visualization version Unicode version |
Description: Membership in the set of integers. (Contributed by NM, 8-Jan-2002.) |
Ref | Expression |
---|---|
elz |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2626 | . . 3 | |
2 | eleq1 2689 | . . 3 | |
3 | negeq 10273 | . . . 4 | |
4 | 3 | eleq1d 2686 | . . 3 |
5 | 1, 2, 4 | 3orbi123d 1398 | . 2 |
6 | df-z 11378 | . 2 | |
7 | 5, 6 | elrab2 3366 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 w3o 1036 wceq 1483 wcel 1990 cr 9935 cc0 9936 cneg 10267 cn 11020 cz 11377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-neg 10269 df-z 11378 |
This theorem is referenced by: nnnegz 11380 zre 11381 elnnz 11387 0z 11388 elznn0nn 11391 elznn0 11392 elznn 11393 znegcl 11412 zeo 11463 addmodlteq 12745 zabsle1 25021 ostthlem1 25316 ostth3 25327 elzdif0 30024 qqhval2lem 30025 |
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