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Mirrors > Home > ILE Home > Th. List > lbzbi | Unicode version |
Description: If a set of reals is bounded below, it is bounded below by an integer. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
lbzbi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1508 | . . 3 | |
2 | nfre1 2476 | . . 3 | |
3 | btwnz 9170 | . . . . . . 7 | |
4 | 3 | simpld 111 | . . . . . 6 |
5 | ssel2 3092 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 | |
6 | zre 9058 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 | |
7 | ltleletr 7846 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 | |
8 | 6, 7 | syl3an1 1249 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |
9 | 8 | expd 256 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |
10 | 9 | 3expia 1183 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 |
11 | 5, 10 | syl5 32 | . . . . . . . . . . . . . . . . . . . . . . . . 25 |
12 | 11 | expdimp 257 | . . . . . . . . . . . . . . . . . . . . . . . 24 |
13 | 12 | com23 78 | . . . . . . . . . . . . . . . . . . . . . . 23 |
14 | 13 | imp 123 | . . . . . . . . . . . . . . . . . . . . . 22 |
15 | 14 | ralrimiv 2504 | . . . . . . . . . . . . . . . . . . . . 21 |
16 | ralim 2491 | . . . . . . . . . . . . . . . . . . . . 21 | |
17 | 15, 16 | syl 14 | . . . . . . . . . . . . . . . . . . . 20 |
18 | 17 | ex 114 | . . . . . . . . . . . . . . . . . . 19 |
19 | 18 | anasss 396 | . . . . . . . . . . . . . . . . . 18 |
20 | 19 | expcom 115 | . . . . . . . . . . . . . . . . 17 |
21 | 20 | com23 78 | . . . . . . . . . . . . . . . 16 |
22 | 21 | imp 123 | . . . . . . . . . . . . . . 15 |
23 | 22 | imdistand 443 | . . . . . . . . . . . . . 14 |
24 | breq1 3932 | . . . . . . . . . . . . . . . 16 | |
25 | 24 | ralbidv 2437 | . . . . . . . . . . . . . . 15 |
26 | 25 | rspcev 2789 | . . . . . . . . . . . . . 14 |
27 | 23, 26 | syl6 33 | . . . . . . . . . . . . 13 |
28 | 27 | ex 114 | . . . . . . . . . . . 12 |
29 | 28 | com23 78 | . . . . . . . . . . 11 |
30 | 29 | ancomsd 267 | . . . . . . . . . 10 |
31 | 30 | expdimp 257 | . . . . . . . . 9 |
32 | 31 | rexlimdv 2548 | . . . . . . . 8 |
33 | 32 | anasss 396 | . . . . . . 7 |
34 | 33 | expcom 115 | . . . . . 6 |
35 | 4, 34 | mpdi 43 | . . . . 5 |
36 | 35 | ex 114 | . . . 4 |
37 | 36 | com23 78 | . . 3 |
38 | 1, 2, 37 | rexlimd 2546 | . 2 |
39 | zssre 9061 | . . 3 | |
40 | ssrexv 3162 | . . 3 | |
41 | 39, 40 | ax-mp 5 | . 2 |
42 | 38, 41 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wcel 1480 wral 2416 wrex 2417 wss 3071 class class class wbr 3929 cr 7619 clt 7800 cle 7801 cz 9054 |
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-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 ax-arch 7739 |
This theorem depends on definitions: df-bi 116 df-3or 963 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-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-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-z 9055 |
This theorem is referenced by: (None) |
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