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Mirrors > Home > ILE Home > Th. List > axpre-ltadd | Unicode version |
Description: Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 7655. (Contributed by NM, 11-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpre-ltadd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 7557 |
. . 3
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2 | elreal 7557 |
. . 3
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3 | elreal 7557 |
. . 3
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4 | breq1 3896 |
. . . 4
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5 | oveq2 5734 |
. . . . 5
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6 | 5 | breq1d 3903 |
. . . 4
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7 | 4, 6 | bibi12d 234 |
. . 3
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8 | breq2 3897 |
. . . 4
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9 | oveq2 5734 |
. . . . 5
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10 | 9 | breq2d 3905 |
. . . 4
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11 | 8, 10 | bibi12d 234 |
. . 3
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12 | oveq1 5733 |
. . . . 5
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13 | oveq1 5733 |
. . . . 5
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14 | 12, 13 | breq12d 3906 |
. . . 4
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15 | 14 | bibi2d 231 |
. . 3
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16 | ltasrg 7507 |
. . . 4
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17 | ltresr 7568 |
. . . . 5
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18 | 17 | a1i 9 |
. . . 4
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19 | simp3 964 |
. . . . . 6
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20 | simp1 962 |
. . . . . 6
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21 | simp2 963 |
. . . . . 6
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22 | addresr 7566 |
. . . . . . 7
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23 | addresr 7566 |
. . . . . . 7
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24 | 22, 23 | breqan12d 3908 |
. . . . . 6
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25 | 19, 20, 19, 21, 24 | syl22anc 1198 |
. . . . 5
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26 | ltresr 7568 |
. . . . 5
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27 | 25, 26 | syl6bb 195 |
. . . 4
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28 | 16, 18, 27 | 3bitr4d 219 |
. . 3
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29 | 1, 2, 3, 7, 11, 15, 28 | 3gencl 2689 |
. 2
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30 | 29 | biimpd 143 |
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 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-iinf 4460 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-eprel 4169 df-id 4173 df-po 4176 df-iso 4177 df-iord 4246 df-on 4248 df-suc 4251 df-iom 4463 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-recs 6154 df-irdg 6219 df-1o 6265 df-2o 6266 df-oadd 6269 df-omul 6270 df-er 6381 df-ec 6383 df-qs 6387 df-ni 7054 df-pli 7055 df-mi 7056 df-lti 7057 df-plpq 7094 df-mpq 7095 df-enq 7097 df-nqqs 7098 df-plqqs 7099 df-mqqs 7100 df-1nqqs 7101 df-rq 7102 df-ltnqqs 7103 df-enq0 7174 df-nq0 7175 df-0nq0 7176 df-plq0 7177 df-mq0 7178 df-inp 7216 df-i1p 7217 df-iplp 7218 df-iltp 7220 df-enr 7463 df-nr 7464 df-plr 7465 df-ltr 7467 df-0r 7468 df-c 7547 df-r 7551 df-add 7552 df-lt 7554 |
This theorem is referenced by: (None) |
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