Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 1461 | . . 3 | |
2 | nfre1 2407 | . . 3 | |
3 | btwnz 8466 | . . . . . . 7 | |
4 | 3 | simpld 110 | . . . . . 6 |
5 | ssel2 2994 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 | |
6 | zre 8355 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 | |
7 | ltleletr 7193 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 | |
8 | 6, 7 | syl3an1 1202 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |
9 | 8 | expd 254 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |
10 | 9 | 3expia 1140 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 |
11 | 5, 10 | syl5 32 | . . . . . . . . . . . . . . . . . . . . . . . . 25 |
12 | 11 | expdimp 255 | . . . . . . . . . . . . . . . . . . . . . . . 24 |
13 | 12 | com23 77 | . . . . . . . . . . . . . . . . . . . . . . 23 |
14 | 13 | imp 122 | . . . . . . . . . . . . . . . . . . . . . 22 |
15 | 14 | ralrimiv 2433 | . . . . . . . . . . . . . . . . . . . . 21 |
16 | ralim 2422 | . . . . . . . . . . . . . . . . . . . . 21 | |
17 | 15, 16 | syl 14 | . . . . . . . . . . . . . . . . . . . 20 |
18 | 17 | ex 113 | . . . . . . . . . . . . . . . . . . 19 |
19 | 18 | anasss 391 | . . . . . . . . . . . . . . . . . 18 |
20 | 19 | expcom 114 | . . . . . . . . . . . . . . . . 17 |
21 | 20 | com23 77 | . . . . . . . . . . . . . . . 16 |
22 | 21 | imp 122 | . . . . . . . . . . . . . . 15 |
23 | 22 | imdistand 435 | . . . . . . . . . . . . . 14 |
24 | breq1 3788 | . . . . . . . . . . . . . . . 16 | |
25 | 24 | ralbidv 2368 | . . . . . . . . . . . . . . 15 |
26 | 25 | rspcev 2701 | . . . . . . . . . . . . . 14 |
27 | 23, 26 | syl6 33 | . . . . . . . . . . . . 13 |
28 | 27 | ex 113 | . . . . . . . . . . . 12 |
29 | 28 | com23 77 | . . . . . . . . . . 11 |
30 | 29 | ancomsd 265 | . . . . . . . . . 10 |
31 | 30 | expdimp 255 | . . . . . . . . 9 |
32 | 31 | rexlimdv 2476 | . . . . . . . 8 |
33 | 32 | anasss 391 | . . . . . . 7 |
34 | 33 | expcom 114 | . . . . . 6 |
35 | 4, 34 | mpdi 42 | . . . . 5 |
36 | 35 | ex 113 | . . . 4 |
37 | 36 | com23 77 | . . 3 |
38 | 1, 2, 37 | rexlimd 2474 | . 2 |
39 | zssre 8358 | . . 3 | |
40 | ssrexv 3059 | . . 3 | |
41 | 39, 40 | ax-mp 7 | . 2 |
42 | 38, 41 | impbid1 140 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wcel 1433 wral 2348 wrex 2349 wss 2973 class class class wbr 3785 cr 6980 clt 7153 cle 7154 cz 8351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 ax-arch 7095 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-z 8352 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |