Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > lbinf | Unicode version |
Description: If a set of reals contains a lower bound, the lower bound is its infimum. (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
Ref | Expression |
---|---|
lbinf | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lttri3 7969 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | lbcl 8832 | . . 3 | |
4 | ssel 3131 | . . . 4 | |
5 | 4 | adantr 274 | . . 3 |
6 | 3, 5 | mpd 13 | . 2 |
7 | 6 | adantr 274 | . . 3 |
8 | ssel2 3132 | . . . 4 | |
9 | 8 | adantlr 469 | . . 3 |
10 | lble 8833 | . . . 4 | |
11 | 10 | 3expa 1192 | . . 3 |
12 | 7, 9, 11 | lensymd 8011 | . 2 |
13 | 2, 6, 3, 12 | infminti 6983 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 wrex 2443 wss 3111 class class class wbr 3976 crio 5791 infcinf 6939 cr 7743 clt 7924 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-sbc 2947 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-iota 5147 df-riota 5792 df-sup 6940 df-inf 6941 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 |
This theorem is referenced by: lbinfcl 8835 lbinfle 8836 |
Copyright terms: Public domain | W3C validator |