Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7158 | . . . . 5 | |
2 | 1 | breqi 3791 | . . . 4 |
3 | brun 3831 | . . . 4 | |
4 | 2, 3 | bitri 182 | . . 3 |
5 | eleq1 2141 | . . . . . . 7 | |
6 | breq1 3788 | . . . . . . 7 | |
7 | 5, 6 | 3anbi13d 1245 | . . . . . 6 |
8 | eleq1 2141 | . . . . . . 7 | |
9 | breq2 3789 | . . . . . . 7 | |
10 | 8, 9 | 3anbi23d 1246 | . . . . . 6 |
11 | eqid 2081 | . . . . . 6 | |
12 | 7, 10, 11 | brabg 4024 | . . . . 5 |
13 | simp3 940 | . . . . 5 | |
14 | 12, 13 | syl6bi 161 | . . . 4 |
15 | brun 3831 | . . . . 5 | |
16 | brxp 4393 | . . . . . . . . . . 11 | |
17 | 16 | simprbi 269 | . . . . . . . . . 10 |
18 | elsni 3416 | . . . . . . . . . 10 | |
19 | 17, 18 | syl 14 | . . . . . . . . 9 |
20 | 19 | a1i 9 | . . . . . . . 8 |
21 | renepnf 7166 | . . . . . . . . 9 | |
22 | 21 | neneqd 2266 | . . . . . . . 8 |
23 | pm2.24 583 | . . . . . . . 8 | |
24 | 20, 22, 23 | syl6ci 1374 | . . . . . . 7 |
25 | 24 | adantl 271 | . . . . . 6 |
26 | brxp 4393 | . . . . . . . . . . 11 | |
27 | 26 | simplbi 268 | . . . . . . . . . 10 |
28 | elsni 3416 | . . . . . . . . . 10 | |
29 | 27, 28 | syl 14 | . . . . . . . . 9 |
30 | 29 | a1i 9 | . . . . . . . 8 |
31 | renemnf 7167 | . . . . . . . . 9 | |
32 | 31 | neneqd 2266 | . . . . . . . 8 |
33 | pm2.24 583 | . . . . . . . 8 | |
34 | 30, 32, 33 | syl6ci 1374 | . . . . . . 7 |
35 | 34 | adantr 270 | . . . . . 6 |
36 | 25, 35 | jaod 669 | . . . . 5 |
37 | 15, 36 | syl5bi 150 | . . . 4 |
38 | 14, 37 | jaod 669 | . . 3 |
39 | 4, 38 | syl5bi 150 | . 2 |
40 | 12 | 3adant3 958 | . . . . . 6 |
41 | 40 | ibir 175 | . . . . 5 |
42 | 41 | orcd 684 | . . . 4 |
43 | 42, 4 | sylibr 132 | . . 3 |
44 | 43 | 3expia 1140 | . 2 |
45 | 39, 44 | impbid 127 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 w3a 919 wceq 1284 wcel 1433 cun 2971 csn 3398 class class class wbr 3785 copab 3838 cxp 4361 cr 6980 cltrr 6985 cpnf 7150 cmnf 7151 clt 7153 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-pnf 7155 df-mnf 7156 df-ltxr 7158 |
This theorem is referenced by: axltirr 7179 axltwlin 7180 axlttrn 7181 axltadd 7182 axapti 7183 axmulgt0 7184 0lt1 7236 recexre 7678 recexgt0 7680 remulext1 7699 arch 8285 caucvgrelemcau 9866 caucvgre 9867 |
Copyright terms: Public domain | W3C validator |