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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 7065 | . . . . 5 | |
2 | 1 | breqi 3770 | . . . 4 |
3 | brun 3810 | . . . 4 | |
4 | 2, 3 | bitri 173 | . . 3 |
5 | eleq1 2100 | . . . . . . 7 | |
6 | breq1 3767 | . . . . . . 7 | |
7 | 5, 6 | 3anbi13d 1209 | . . . . . 6 |
8 | eleq1 2100 | . . . . . . 7 | |
9 | breq2 3768 | . . . . . . 7 | |
10 | 8, 9 | 3anbi23d 1210 | . . . . . 6 |
11 | eqid 2040 | . . . . . 6 | |
12 | 7, 10, 11 | brabg 4006 | . . . . 5 |
13 | simp3 906 | . . . . 5 | |
14 | 12, 13 | syl6bi 152 | . . . 4 |
15 | brun 3810 | . . . . 5 | |
16 | brxp 4375 | . . . . . . . . . . 11 | |
17 | 16 | simprbi 260 | . . . . . . . . . 10 |
18 | elsni 3393 | . . . . . . . . . 10 | |
19 | 17, 18 | syl 14 | . . . . . . . . 9 |
20 | 19 | a1i 9 | . . . . . . . 8 |
21 | renepnf 7073 | . . . . . . . . 9 | |
22 | 21 | neneqd 2226 | . . . . . . . 8 |
23 | pm2.24 551 | . . . . . . . 8 | |
24 | 20, 22, 23 | syl6ci 1334 | . . . . . . 7 |
25 | 24 | adantl 262 | . . . . . 6 |
26 | brxp 4375 | . . . . . . . . . . 11 | |
27 | 26 | simplbi 259 | . . . . . . . . . 10 |
28 | elsni 3393 | . . . . . . . . . 10 | |
29 | 27, 28 | syl 14 | . . . . . . . . 9 |
30 | 29 | a1i 9 | . . . . . . . 8 |
31 | renemnf 7074 | . . . . . . . . 9 | |
32 | 31 | neneqd 2226 | . . . . . . . 8 |
33 | pm2.24 551 | . . . . . . . 8 | |
34 | 30, 32, 33 | syl6ci 1334 | . . . . . . 7 |
35 | 34 | adantr 261 | . . . . . 6 |
36 | 25, 35 | jaod 637 | . . . . 5 |
37 | 15, 36 | syl5bi 141 | . . . 4 |
38 | 14, 37 | jaod 637 | . . 3 |
39 | 4, 38 | syl5bi 141 | . 2 |
40 | 12 | 3adant3 924 | . . . . . 6 |
41 | 40 | ibir 166 | . . . . 5 |
42 | 41 | orcd 652 | . . . 4 |
43 | 42, 4 | sylibr 137 | . . 3 |
44 | 43 | 3expia 1106 | . 2 |
45 | 39, 44 | impbid 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 wo 629 w3a 885 wceq 1243 wcel 1393 cun 2915 csn 3375 class class class wbr 3764 copab 3817 cxp 4343 cr 6888 cltrr 6893 cpnf 7057 cmnf 7058 clt 7060 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-pnf 7062 df-mnf 7063 df-ltxr 7065 |
This theorem is referenced by: axltirr 7086 axltwlin 7087 axlttrn 7088 axltadd 7089 axapti 7090 axmulgt0 7091 0lt1 7141 recexre 7569 recexgt0 7571 remulext1 7590 arch 8178 caucvgrelemcau 9579 caucvgre 9580 |
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