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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 1493 | . . 3 | |
2 | nfre1 2453 | . . 3 | |
3 | btwnz 9138 | . . . . . . 7 | |
4 | 3 | simpld 111 | . . . . . 6 |
5 | ssel2 3062 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 | |
6 | zre 9026 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 | |
7 | ltleletr 7814 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 | |
8 | 6, 7 | syl3an1 1234 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |
9 | 8 | expd 256 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |
10 | 9 | 3expia 1168 | . . . . . . . . . . . . . . . . . . . . . . . . . 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 2481 | . . . . . . . . . . . . . . . . . . . . 21 |
16 | ralim 2468 | . . . . . . . . . . . . . . . . . . . . 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 3902 | . . . . . . . . . . . . . . . 16 | |
25 | 24 | ralbidv 2414 | . . . . . . . . . . . . . . 15 |
26 | 25 | rspcev 2763 | . . . . . . . . . . . . . 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 2525 | . . . . . . . 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 2523 | . 2 |
39 | zssre 9029 | . . 3 | |
40 | ssrexv 3132 | . . 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 947 wcel 1465 wral 2393 wrex 2394 wss 3041 class class class wbr 3899 cr 7587 clt 7768 cle 7769 cz 9022 |
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-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 ax-arch 7707 |
This theorem depends on definitions: df-bi 116 df-3or 948 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-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-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-z 9023 |
This theorem is referenced by: (None) |
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