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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 7734. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpre-lttrn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 7636 | . 2 | |
2 | elreal 7636 | . 2 | |
3 | elreal 7636 | . 2 | |
4 | breq1 3932 | . . . 4 | |
5 | 4 | anbi1d 460 | . . 3 |
6 | breq1 3932 | . . 3 | |
7 | 5, 6 | imbi12d 233 | . 2 |
8 | breq2 3933 | . . . 4 | |
9 | breq1 3932 | . . . 4 | |
10 | 8, 9 | anbi12d 464 | . . 3 |
11 | 10 | imbi1d 230 | . 2 |
12 | breq2 3933 | . . . 4 | |
13 | 12 | anbi2d 459 | . . 3 |
14 | breq2 3933 | . . 3 | |
15 | 13, 14 | imbi12d 233 | . 2 |
16 | ltresr 7647 | . . . . 5 | |
17 | ltresr 7647 | . . . . 5 | |
18 | ltsosr 7572 | . . . . . 6 | |
19 | ltrelsr 7546 | . . . . . 6 | |
20 | 18, 19 | sotri 4934 | . . . . 5 |
21 | 16, 17, 20 | syl2anb 289 | . . . 4 |
22 | ltresr 7647 | . . . 4 | |
23 | 21, 22 | sylibr 133 | . . 3 |
24 | 23 | a1i 9 | . 2 |
25 | 1, 2, 3, 7, 11, 15, 24 | 3gencl 2720 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 cop 3530 class class class wbr 3929 cnr 7105 c0r 7106 cltr 7111 cr 7619 cltrr 7624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-i1p 7275 df-iplp 7276 df-iltp 7278 df-enr 7534 df-nr 7535 df-ltr 7538 df-0r 7539 df-r 7630 df-lt 7633 |
This theorem is referenced by: (None) |
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