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Mirrors > Home > ILE Home > Th. List > axltadd | Unicode version |
Description: Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7000 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
Ref | Expression |
---|---|
axltadd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pre-ltadd 7000 | . 2 | |
2 | ltxrlt 7085 | . . 3 | |
3 | 2 | 3adant3 924 | . 2 |
4 | readdcl 7007 | . . . . 5 | |
5 | readdcl 7007 | . . . . 5 | |
6 | ltxrlt 7085 | . . . . 5 | |
7 | 4, 5, 6 | syl2an 273 | . . . 4 |
8 | 7 | 3impdi 1190 | . . 3 |
9 | 8 | 3coml 1111 | . 2 |
10 | 1, 3, 9 | 3imtr4d 192 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wcel 1393 class class class wbr 3764 (class class class)co 5512 cr 6888 caddc 6892 cltrr 6893 clt 7060 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-addrcl 6981 ax-pre-ltadd 7000 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-pnf 7062 df-mnf 7063 df-ltxr 7065 |
This theorem is referenced by: ltadd2 7416 nnge1 7937 |
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