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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 7522 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
Ref | Expression |
---|---|
axltadd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pre-ltadd 7522 |
. 2
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2 | ltxrlt 7613 |
. . 3
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3 | 2 | 3adant3 964 |
. 2
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4 | readdcl 7529 |
. . . . 5
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5 | readdcl 7529 |
. . . . 5
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6 | ltxrlt 7613 |
. . . . 5
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7 | 4, 5, 6 | syl2an 284 |
. . . 4
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8 | 7 | 3impdi 1230 |
. . 3
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9 | 8 | 3coml 1151 |
. 2
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10 | 1, 3, 9 | 3imtr4d 202 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-addrcl 7503 ax-pre-ltadd 7522 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-xp 4458 df-pnf 7585 df-mnf 7586 df-ltxr 7588 |
This theorem is referenced by: ltadd2 7958 nnge1 8506 ltoddhalfle 11232 |
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