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Mirrors > Home > ILE Home > Th. List > ltxrlt | Unicode version |
Description: The standard less-than and the extended real less-than are identical when restricted to the non-extended reals . (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
ltxrlt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ltxr 7829 | . . . . 5 | |
2 | 1 | breqi 3943 | . . . 4 |
3 | brun 3987 | . . . 4 | |
4 | 2, 3 | bitri 183 | . . 3 |
5 | eleq1 2203 | . . . . . . 7 | |
6 | breq1 3940 | . . . . . . 7 | |
7 | 5, 6 | 3anbi13d 1293 | . . . . . 6 |
8 | eleq1 2203 | . . . . . . 7 | |
9 | breq2 3941 | . . . . . . 7 | |
10 | 8, 9 | 3anbi23d 1294 | . . . . . 6 |
11 | eqid 2140 | . . . . . 6 | |
12 | 7, 10, 11 | brabg 4199 | . . . . 5 |
13 | simp3 984 | . . . . 5 | |
14 | 12, 13 | syl6bi 162 | . . . 4 |
15 | brun 3987 | . . . . 5 | |
16 | brxp 4578 | . . . . . . . . . . 11 | |
17 | 16 | simprbi 273 | . . . . . . . . . 10 |
18 | elsni 3550 | . . . . . . . . . 10 | |
19 | 17, 18 | syl 14 | . . . . . . . . 9 |
20 | 19 | a1i 9 | . . . . . . . 8 |
21 | renepnf 7837 | . . . . . . . . 9 | |
22 | 21 | neneqd 2330 | . . . . . . . 8 |
23 | pm2.24 611 | . . . . . . . 8 | |
24 | 20, 22, 23 | syl6ci 1422 | . . . . . . 7 |
25 | 24 | adantl 275 | . . . . . 6 |
26 | brxp 4578 | . . . . . . . . . . 11 | |
27 | 26 | simplbi 272 | . . . . . . . . . 10 |
28 | elsni 3550 | . . . . . . . . . 10 | |
29 | 27, 28 | syl 14 | . . . . . . . . 9 |
30 | 29 | a1i 9 | . . . . . . . 8 |
31 | renemnf 7838 | . . . . . . . . 9 | |
32 | 31 | neneqd 2330 | . . . . . . . 8 |
33 | pm2.24 611 | . . . . . . . 8 | |
34 | 30, 32, 33 | syl6ci 1422 | . . . . . . 7 |
35 | 34 | adantr 274 | . . . . . 6 |
36 | 25, 35 | jaod 707 | . . . . 5 |
37 | 15, 36 | syl5bi 151 | . . . 4 |
38 | 14, 37 | jaod 707 | . . 3 |
39 | 4, 38 | syl5bi 151 | . 2 |
40 | 12 | 3adant3 1002 | . . . . . 6 |
41 | 40 | ibir 176 | . . . . 5 |
42 | 41 | orcd 723 | . . . 4 |
43 | 42, 4 | sylibr 133 | . . 3 |
44 | 43 | 3expia 1184 | . 2 |
45 | 39, 44 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 w3a 963 wceq 1332 wcel 1481 cun 3074 csn 3532 class class class wbr 3937 copab 3996 cxp 4545 cr 7643 cltrr 7648 cpnf 7821 cmnf 7822 clt 7824 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-pnf 7826 df-mnf 7827 df-ltxr 7829 |
This theorem is referenced by: axltirr 7855 axltwlin 7856 axlttrn 7857 axltadd 7858 axapti 7859 axmulgt0 7860 axsuploc 7861 0lt1 7913 recexre 8364 recexgt0 8366 remulext1 8385 arch 8998 caucvgrelemcau 10784 caucvgre 10785 |
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