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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 3980 | . . . . . . . . 9 | |
2 | 1 | rspcv 2821 | . . . . . . . 8 |
3 | breq2 3980 | . . . . . . . . 9 | |
4 | 3 | rspcv 2821 | . . . . . . . 8 |
5 | 2, 4 | im2anan9r 589 | . . . . . . 7 |
6 | ssel 3131 | . . . . . . . . . . . 12 | |
7 | ssel 3131 | . . . . . . . . . . . 12 | |
8 | 6, 7 | anim12d 333 | . . . . . . . . . . 11 |
9 | 8 | impcom 124 | . . . . . . . . . 10 |
10 | letri3 7970 | . . . . . . . . . 10 | |
11 | 9, 10 | syl 14 | . . . . . . . . 9 |
12 | 11 | exbiri 380 | . . . . . . . 8 |
13 | 12 | com23 78 | . . . . . . 7 |
14 | 5, 13 | syld 45 | . . . . . 6 |
15 | 14 | com3r 79 | . . . . 5 |
16 | 15 | ralrimivv 2545 | . . . 4 |
17 | 16 | anim2i 340 | . . 3 |
18 | 17 | ancoms 266 | . 2 |
19 | breq1 3979 | . . . 4 | |
20 | 19 | ralbidv 2464 | . . 3 |
21 | 20 | reu4 2915 | . 2 |
22 | 18, 21 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2135 wral 2442 wrex 2443 wreu 2444 wss 3111 class class class wbr 3976 cr 7743 cle 7925 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-pre-ltirr 7856 ax-pre-apti 7859 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 |
This theorem is referenced by: lbcl 8832 lble 8833 |
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