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Mirrors > Home > ILE Home > Th. List > rexneg | Unicode version |
Description: Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
rexneg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xneg 8843 | . 2 | |
2 | renepnf 7166 | . . . 4 | |
3 | ifnefalse 3362 | . . . 4 | |
4 | 2, 3 | syl 14 | . . 3 |
5 | renemnf 7167 | . . . 4 | |
6 | ifnefalse 3362 | . . . 4 | |
7 | 5, 6 | syl 14 | . . 3 |
8 | 4, 7 | eqtrd 2113 | . 2 |
9 | 1, 8 | syl5eq 2125 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1284 wcel 1433 wne 2245 cif 3351 cr 6980 cpnf 7150 cmnf 7151 cneg 7280 cxne 8840 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-pnf 7155 df-mnf 7156 df-xneg 8843 |
This theorem is referenced by: xneg0 8898 xnegcl 8899 xnegneg 8900 xltnegi 8902 |
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