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Mirrors > Home > ILE Home > Th. List > lble | Unicode version |
Description: If a set of reals contains a lower bound, the lower bound is less than or equal to all members of the set. (Contributed by NM, 9-Oct-2005.) (Proof shortened by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
lble |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lbreu 8671 | . . . . 5 | |
2 | nfcv 2258 | . . . . . . 7 | |
3 | nfriota1 5705 | . . . . . . . 8 | |
4 | nfcv 2258 | . . . . . . . 8 | |
5 | nfcv 2258 | . . . . . . . 8 | |
6 | 3, 4, 5 | nfbr 3944 | . . . . . . 7 |
7 | 2, 6 | nfralxy 2448 | . . . . . 6 |
8 | eqid 2117 | . . . . . 6 | |
9 | nfra1 2443 | . . . . . . . . 9 | |
10 | nfcv 2258 | . . . . . . . . 9 | |
11 | 9, 10 | nfriota 5707 | . . . . . . . 8 |
12 | 11 | nfeq2 2270 | . . . . . . 7 |
13 | breq1 3902 | . . . . . . 7 | |
14 | 12, 13 | ralbid 2412 | . . . . . 6 |
15 | 7, 8, 14 | riotaprop 5721 | . . . . 5 |
16 | 1, 15 | syl 14 | . . . 4 |
17 | 16 | simprd 113 | . . 3 |
18 | nfcv 2258 | . . . . 5 | |
19 | nfcv 2258 | . . . . 5 | |
20 | 11, 18, 19 | nfbr 3944 | . . . 4 |
21 | breq2 3903 | . . . 4 | |
22 | 20, 21 | rspc 2757 | . . 3 |
23 | 17, 22 | mpan9 279 | . 2 |
24 | 23 | 3impa 1161 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 wral 2393 wrex 2394 wreu 2395 wss 3041 class class class wbr 3899 crio 5697 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-sbc 2883 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-iota 5058 df-riota 5698 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 |
This theorem is referenced by: lbinf 8674 lbinfle 8676 |
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