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Mirrors > Home > ILE Home > Th. List > lbinfcl | Unicode version |
Description: If a set of reals contains a lower bound, it contains its infimum. (Contributed by NM, 11-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
Ref | Expression |
---|---|
lbinfcl | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lbinf 8711 | . 2 inf | |
2 | lbcl 8709 | . 2 | |
3 | 1, 2 | eqeltrd 2216 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1480 wral 2416 wrex 2417 wss 3071 class class class wbr 3929 crio 5729 infcinf 6870 cr 7624 clt 7805 cle 7806 |
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 7716 ax-resscn 7717 ax-pre-ltirr 7737 ax-pre-apti 7740 |
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-sbc 2910 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-iota 5088 df-riota 5730 df-sup 6871 df-inf 6872 df-pnf 7807 df-mnf 7808 df-xr 7809 df-ltxr 7810 df-le 7811 |
This theorem is referenced by: (None) |
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