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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 7917 | . . . . 5 | |
2 | 1 | breqi 3971 | . . . 4 |
3 | brun 4015 | . . . 4 | |
4 | 2, 3 | bitri 183 | . . 3 |
5 | eleq1 2220 | . . . . . . 7 | |
6 | breq1 3968 | . . . . . . 7 | |
7 | 5, 6 | 3anbi13d 1296 | . . . . . 6 |
8 | eleq1 2220 | . . . . . . 7 | |
9 | breq2 3969 | . . . . . . 7 | |
10 | 8, 9 | 3anbi23d 1297 | . . . . . 6 |
11 | eqid 2157 | . . . . . 6 | |
12 | 7, 10, 11 | brabg 4229 | . . . . 5 |
13 | simp3 984 | . . . . 5 | |
14 | 12, 13 | syl6bi 162 | . . . 4 |
15 | brun 4015 | . . . . 5 | |
16 | brxp 4617 | . . . . . . . . . . 11 | |
17 | 16 | simprbi 273 | . . . . . . . . . 10 |
18 | elsni 3578 | . . . . . . . . . 10 | |
19 | 17, 18 | syl 14 | . . . . . . . . 9 |
20 | 19 | a1i 9 | . . . . . . . 8 |
21 | renepnf 7925 | . . . . . . . . 9 | |
22 | 21 | neneqd 2348 | . . . . . . . 8 |
23 | pm2.24 611 | . . . . . . . 8 | |
24 | 20, 22, 23 | syl6ci 1425 | . . . . . . 7 |
25 | 24 | adantl 275 | . . . . . 6 |
26 | brxp 4617 | . . . . . . . . . . 11 | |
27 | 26 | simplbi 272 | . . . . . . . . . 10 |
28 | elsni 3578 | . . . . . . . . . 10 | |
29 | 27, 28 | syl 14 | . . . . . . . . 9 |
30 | 29 | a1i 9 | . . . . . . . 8 |
31 | renemnf 7926 | . . . . . . . . 9 | |
32 | 31 | neneqd 2348 | . . . . . . . 8 |
33 | pm2.24 611 | . . . . . . . 8 | |
34 | 30, 32, 33 | syl6ci 1425 | . . . . . . 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 1187 | . 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 1335 wcel 2128 cun 3100 csn 3560 class class class wbr 3965 copab 4024 cxp 4584 cr 7731 cltrr 7736 cpnf 7909 cmnf 7910 clt 7912 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-xp 4592 df-pnf 7914 df-mnf 7915 df-ltxr 7917 |
This theorem is referenced by: axltirr 7944 axltwlin 7945 axlttrn 7946 axltadd 7947 axapti 7948 axmulgt0 7949 axsuploc 7950 0lt1 8002 recexre 8453 recexgt0 8455 remulext1 8474 arch 9087 caucvgrelemcau 10880 caucvgre 10881 |
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