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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 2500 | . . 3 | |
3 | btwnz 9283 | . . . . . . 7 | |
4 | 3 | simpld 111 | . . . . . 6 |
5 | ssel2 3123 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 | |
6 | zre 9171 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 | |
7 | ltleletr 7959 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 | |
8 | 6, 7 | syl3an1 1253 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |
9 | 8 | expd 256 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |
10 | 9 | 3expia 1187 | . . . . . . . . . . . . . . . . . . . . . . . . . 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 2529 | . . . . . . . . . . . . . . . . . . . . 21 |
16 | ralim 2516 | . . . . . . . . . . . . . . . . . . . . 21 | |
17 | 15, 16 | syl 14 | . . . . . . . . . . . . . . . . . . . 20 |
18 | 17 | ex 114 | . . . . . . . . . . . . . . . . . . 19 |
19 | 18 | anasss 397 | . . . . . . . . . . . . . . . . . 18 |
20 | 19 | expcom 115 | . . . . . . . . . . . . . . . . 17 |
21 | 20 | com23 78 | . . . . . . . . . . . . . . . 16 |
22 | 21 | imp 123 | . . . . . . . . . . . . . . 15 |
23 | 22 | imdistand 444 | . . . . . . . . . . . . . 14 |
24 | breq1 3968 | . . . . . . . . . . . . . . . 16 | |
25 | 24 | ralbidv 2457 | . . . . . . . . . . . . . . 15 |
26 | 25 | rspcev 2816 | . . . . . . . . . . . . . 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 2573 | . . . . . . . 8 |
33 | 32 | anasss 397 | . . . . . . 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 2571 | . 2 |
39 | zssre 9174 | . . 3 | |
40 | ssrexv 3193 | . . 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 963 wcel 2128 wral 2435 wrex 2436 wss 3102 class class class wbr 3965 cr 7731 clt 7912 cle 7913 cz 9167 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-addcom 7832 ax-addass 7834 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-0id 7840 ax-rnegex 7841 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-ltadd 7848 ax-arch 7851 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-inn 8834 df-z 9168 |
This theorem is referenced by: (None) |
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