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Mirrors > Home > ILE Home > Th. List > lbreu | Unicode version |
Description: If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.) |
Ref | Expression |
---|---|
lbreu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 3903 | . . . . . . . . 9 | |
2 | 1 | rspcv 2759 | . . . . . . . 8 |
3 | breq2 3903 | . . . . . . . . 9 | |
4 | 3 | rspcv 2759 | . . . . . . . 8 |
5 | 2, 4 | im2anan9r 573 | . . . . . . 7 |
6 | ssel 3061 | . . . . . . . . . . . 12 | |
7 | ssel 3061 | . . . . . . . . . . . 12 | |
8 | 6, 7 | anim12d 333 | . . . . . . . . . . 11 |
9 | 8 | impcom 124 | . . . . . . . . . 10 |
10 | letri3 7813 | . . . . . . . . . 10 | |
11 | 9, 10 | syl 14 | . . . . . . . . 9 |
12 | 11 | exbiri 379 | . . . . . . . 8 |
13 | 12 | com23 78 | . . . . . . 7 |
14 | 5, 13 | syld 45 | . . . . . 6 |
15 | 14 | com3r 79 | . . . . 5 |
16 | 15 | ralrimivv 2490 | . . . 4 |
17 | 16 | anim2i 339 | . . 3 |
18 | 17 | ancoms 266 | . 2 |
19 | breq1 3902 | . . . 4 | |
20 | 19 | ralbidv 2414 | . . 3 |
21 | 20 | reu4 2851 | . 2 |
22 | 18, 21 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1465 wral 2393 wrex 2394 wreu 2395 wss 3041 class class class wbr 3899 cr 7587 cle 7769 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltirr 7700 ax-pre-apti 7703 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 |
This theorem is referenced by: lbcl 8672 lble 8673 |
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