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Mirrors > Home > ILE Home > Th. List > exbtwnz | Unicode version |
Description: If a real number is between an integer and its successor, there is a unique greatest integer less than or equal to the real number. (Contributed by Jim Kingdon, 10-May-2022.) |
Ref | Expression |
---|---|
exbtwnz.ex |
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exbtwnz.a |
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Ref | Expression |
---|---|
exbtwnz |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exbtwnz.ex |
. 2
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2 | simplrl 505 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 2 | zred 9025 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | exbtwnz.a |
. . . . . . . . 9
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5 | 4 | ad2antrr 475 |
. . . . . . . 8
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6 | simplrr 506 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 6 | zred 9025 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
8 | 1red 7653 |
. . . . . . . . 9
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9 | 7, 8 | readdcld 7667 |
. . . . . . . 8
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10 | simprll 507 |
. . . . . . . 8
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11 | simprrr 510 |
. . . . . . . 8
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12 | 3, 5, 9, 10, 11 | lelttrd 7758 |
. . . . . . 7
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13 | zleltp1 8961 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | 2, 6, 13 | syl2anc 406 |
. . . . . . 7
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15 | 12, 14 | mpbird 166 |
. . . . . 6
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16 | 3, 8 | readdcld 7667 |
. . . . . . . 8
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17 | simprrl 509 |
. . . . . . . 8
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18 | simprlr 508 |
. . . . . . . 8
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19 | 7, 5, 16, 17, 18 | lelttrd 7758 |
. . . . . . 7
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20 | zleltp1 8961 |
. . . . . . . 8
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21 | 6, 2, 20 | syl2anc 406 |
. . . . . . 7
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22 | 19, 21 | mpbird 166 |
. . . . . 6
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23 | 3, 7 | letri3d 7750 |
. . . . . 6
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24 | 15, 22, 23 | mpbir2and 896 |
. . . . 5
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25 | 24 | ex 114 |
. . . 4
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26 | 25 | ralrimivva 2473 |
. . 3
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27 | breq1 3878 |
. . . . 5
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28 | oveq1 5713 |
. . . . . 6
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29 | 28 | breq2d 3887 |
. . . . 5
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30 | 27, 29 | anbi12d 460 |
. . . 4
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31 | 30 | rmo4 2830 |
. . 3
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32 | 26, 31 | sylibr 133 |
. 2
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33 | reu5 2601 |
. 2
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34 | 1, 32, 33 | sylanbrc 411 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-addcom 7595 ax-addass 7597 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-0id 7603 ax-rnegex 7604 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-opab 3930 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-inn 8579 df-n0 8830 df-z 8907 |
This theorem is referenced by: qbtwnz 9870 apbtwnz 9888 |
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