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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 1521 | . . 3 | |
2 | nfre1 2513 | . . 3 | |
3 | btwnz 9331 | . . . . . . 7 | |
4 | 3 | simpld 111 | . . . . . 6 |
5 | ssel2 3142 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 | |
6 | zre 9216 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 | |
7 | ltleletr 8001 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 | |
8 | 6, 7 | syl3an1 1266 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |
9 | 8 | expd 256 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |
10 | 9 | 3expia 1200 | . . . . . . . . . . . . . . . . . . . . . . . . . 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 2542 | . . . . . . . . . . . . . . . . . . . . 21 |
16 | ralim 2529 | . . . . . . . . . . . . . . . . . . . . 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 445 | . . . . . . . . . . . . . 14 |
24 | breq1 3992 | . . . . . . . . . . . . . . . 16 | |
25 | 24 | ralbidv 2470 | . . . . . . . . . . . . . . 15 |
26 | 25 | rspcev 2834 | . . . . . . . . . . . . . 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 2586 | . . . . . . . 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 2584 | . 2 |
39 | zssre 9219 | . . 3 | |
40 | ssrexv 3212 | . . 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 973 wcel 2141 wral 2448 wrex 2449 wss 3121 class class class wbr 3989 cr 7773 clt 7954 cle 7955 cz 9212 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 ax-arch 7893 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-z 9213 |
This theorem is referenced by: (None) |
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