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Mirrors > Home > ILE Home > Th. List > axpre-lttrn | Unicode version |
Description: Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 7954. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpre-lttrn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 7856 |
. 2
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2 | elreal 7856 |
. 2
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3 | elreal 7856 |
. 2
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4 | breq1 4021 |
. . . 4
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5 | 4 | anbi1d 465 |
. . 3
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6 | breq1 4021 |
. . 3
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7 | 5, 6 | imbi12d 234 |
. 2
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8 | breq2 4022 |
. . . 4
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9 | breq1 4021 |
. . . 4
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10 | 8, 9 | anbi12d 473 |
. . 3
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11 | 10 | imbi1d 231 |
. 2
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12 | breq2 4022 |
. . . 4
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13 | 12 | anbi2d 464 |
. . 3
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14 | breq2 4022 |
. . 3
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15 | 13, 14 | imbi12d 234 |
. 2
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16 | ltresr 7867 |
. . . . 5
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17 | ltresr 7867 |
. . . . 5
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18 | ltsosr 7792 |
. . . . . 6
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19 | ltrelsr 7766 |
. . . . . 6
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20 | 18, 19 | sotri 5042 |
. . . . 5
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21 | 16, 17, 20 | syl2anb 291 |
. . . 4
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22 | ltresr 7867 |
. . . 4
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23 | 21, 22 | sylibr 134 |
. . 3
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24 | 23 | a1i 9 |
. 2
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25 | 1, 2, 3, 7, 11, 15, 24 | 3gencl 2786 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4307 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-recs 6329 df-irdg 6394 df-1o 6440 df-2o 6441 df-oadd 6444 df-omul 6445 df-er 6558 df-ec 6560 df-qs 6564 df-ni 7332 df-pli 7333 df-mi 7334 df-lti 7335 df-plpq 7372 df-mpq 7373 df-enq 7375 df-nqqs 7376 df-plqqs 7377 df-mqqs 7378 df-1nqqs 7379 df-rq 7380 df-ltnqqs 7381 df-enq0 7452 df-nq0 7453 df-0nq0 7454 df-plq0 7455 df-mq0 7456 df-inp 7494 df-i1p 7495 df-iplp 7496 df-iltp 7498 df-enr 7754 df-nr 7755 df-ltr 7758 df-0r 7759 df-r 7850 df-lt 7853 |
This theorem is referenced by: (None) |
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