Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3933 | . . . . . . . . 9 | |
2 | 1 | rspcv 2785 | . . . . . . . 8 |
3 | breq2 3933 | . . . . . . . . 9 | |
4 | 3 | rspcv 2785 | . . . . . . . 8 |
5 | 2, 4 | im2anan9r 588 | . . . . . . 7 |
6 | ssel 3091 | . . . . . . . . . . . 12 | |
7 | ssel 3091 | . . . . . . . . . . . 12 | |
8 | 6, 7 | anim12d 333 | . . . . . . . . . . 11 |
9 | 8 | impcom 124 | . . . . . . . . . 10 |
10 | letri3 7845 | . . . . . . . . . 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 2513 | . . . 4 |
17 | 16 | anim2i 339 | . . 3 |
18 | 17 | ancoms 266 | . 2 |
19 | breq1 3932 | . . . 4 | |
20 | 19 | ralbidv 2437 | . . 3 |
21 | 20 | reu4 2878 | . 2 |
22 | 18, 21 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1480 wral 2416 wrex 2417 wreu 2418 wss 3071 class class class wbr 3929 cr 7619 cle 7801 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732 ax-pre-apti 7735 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 |
This theorem is referenced by: lbcl 8704 lble 8705 |
Copyright terms: Public domain | W3C validator |