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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 7959 | . . . . 5 | |
2 | 1 | breqi 3995 | . . . 4 |
3 | brun 4040 | . . . 4 | |
4 | 2, 3 | bitri 183 | . . 3 |
5 | eleq1 2233 | . . . . . . 7 | |
6 | breq1 3992 | . . . . . . 7 | |
7 | 5, 6 | 3anbi13d 1309 | . . . . . 6 |
8 | eleq1 2233 | . . . . . . 7 | |
9 | breq2 3993 | . . . . . . 7 | |
10 | 8, 9 | 3anbi23d 1310 | . . . . . 6 |
11 | eqid 2170 | . . . . . 6 | |
12 | 7, 10, 11 | brabg 4254 | . . . . 5 |
13 | simp3 994 | . . . . 5 | |
14 | 12, 13 | syl6bi 162 | . . . 4 |
15 | brun 4040 | . . . . 5 | |
16 | brxp 4642 | . . . . . . . . . . 11 | |
17 | 16 | simprbi 273 | . . . . . . . . . 10 |
18 | elsni 3601 | . . . . . . . . . 10 | |
19 | 17, 18 | syl 14 | . . . . . . . . 9 |
20 | 19 | a1i 9 | . . . . . . . 8 |
21 | renepnf 7967 | . . . . . . . . 9 | |
22 | 21 | neneqd 2361 | . . . . . . . 8 |
23 | pm2.24 616 | . . . . . . . 8 | |
24 | 20, 22, 23 | syl6ci 1438 | . . . . . . 7 |
25 | 24 | adantl 275 | . . . . . 6 |
26 | brxp 4642 | . . . . . . . . . . 11 | |
27 | 26 | simplbi 272 | . . . . . . . . . 10 |
28 | elsni 3601 | . . . . . . . . . 10 | |
29 | 27, 28 | syl 14 | . . . . . . . . 9 |
30 | 29 | a1i 9 | . . . . . . . 8 |
31 | renemnf 7968 | . . . . . . . . 9 | |
32 | 31 | neneqd 2361 | . . . . . . . 8 |
33 | pm2.24 616 | . . . . . . . 8 | |
34 | 30, 32, 33 | syl6ci 1438 | . . . . . . 7 |
35 | 34 | adantr 274 | . . . . . 6 |
36 | 25, 35 | jaod 712 | . . . . 5 |
37 | 15, 36 | syl5bi 151 | . . . 4 |
38 | 14, 37 | jaod 712 | . . 3 |
39 | 4, 38 | syl5bi 151 | . 2 |
40 | 12 | 3adant3 1012 | . . . . . 6 |
41 | 40 | ibir 176 | . . . . 5 |
42 | 41 | orcd 728 | . . . 4 |
43 | 42, 4 | sylibr 133 | . . 3 |
44 | 43 | 3expia 1200 | . 2 |
45 | 39, 44 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 w3a 973 wceq 1348 wcel 2141 cun 3119 csn 3583 class class class wbr 3989 copab 4049 cxp 4609 cr 7773 cltrr 7778 cpnf 7951 cmnf 7952 clt 7954 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-pnf 7956 df-mnf 7957 df-ltxr 7959 |
This theorem is referenced by: axltirr 7986 axltwlin 7987 axlttrn 7988 axltadd 7989 axapti 7990 axmulgt0 7991 axsuploc 7992 0lt1 8046 recexre 8497 recexgt0 8499 remulext1 8518 arch 9132 caucvgrelemcau 10944 caucvgre 10945 |
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