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Mirrors > Home > ILE Home > Th. List > exbtwnzlemex | Unicode version |
Description: Existence of an integer
so that a given real number is between the
integer and its successor. The real number must satisfy the
![]() ![]() ![]() ![]() ![]() ![]() ![]()
The proof starts by finding two integers which are less than and greater
than |
Ref | Expression |
---|---|
exbtwnzlemex.a |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
exbtwnzlemex.tri |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Ref | Expression |
---|---|
exbtwnzlemex |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exbtwnzlemex.a |
. . . 4
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2 | btwnz 9194 |
. . . 4
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3 | 1, 2 | syl 14 |
. . 3
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4 | reeanv 2603 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 3, 4 | sylibr 133 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | simplrl 525 |
. . . . . 6
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7 | 6 | zred 9197 |
. . . . . . 7
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8 | 1 | ad2antrr 480 |
. . . . . . 7
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9 | simprl 521 |
. . . . . . 7
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10 | 7, 8, 9 | ltled 7905 |
. . . . . 6
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11 | simprr 522 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 6 | zcnd 9198 |
. . . . . . . 8
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13 | simplrr 526 |
. . . . . . . . 9
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14 | 13 | zcnd 9198 |
. . . . . . . 8
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15 | 12, 14 | pncan3d 8100 |
. . . . . . 7
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16 | 11, 15 | breqtrrd 3964 |
. . . . . 6
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17 | breq1 3940 |
. . . . . . . 8
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18 | oveq1 5789 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 18 | breq2d 3949 |
. . . . . . . 8
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20 | 17, 19 | anbi12d 465 |
. . . . . . 7
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21 | 20 | rspcev 2793 |
. . . . . 6
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22 | 6, 10, 16, 21 | syl12anc 1215 |
. . . . 5
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23 | 13 | zred 9197 |
. . . . . . . 8
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24 | 7, 8, 23, 9, 11 | lttrd 7912 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | znnsub 9129 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 25 | ad2antlr 481 |
. . . . . . 7
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27 | 24, 26 | mpbid 146 |
. . . . . 6
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28 | exbtwnzlemex.tri |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 28 | ralrimiva 2508 |
. . . . . . . . 9
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30 | breq1 3940 |
. . . . . . . . . . 11
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31 | breq2 3941 |
. . . . . . . . . . 11
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32 | 30, 31 | orbi12d 783 |
. . . . . . . . . 10
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33 | 32 | cbvralv 2657 |
. . . . . . . . 9
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34 | 29, 33 | sylib 121 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 34 | ad2antrr 480 |
. . . . . . 7
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36 | 35 | r19.21bi 2523 |
. . . . . 6
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37 | 27, 8, 36 | exbtwnzlemshrink 10057 |
. . . . 5
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38 | 22, 37 | mpdan 418 |
. . . 4
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39 | 38 | ex 114 |
. . 3
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40 | 39 | rexlimdvva 2560 |
. 2
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41 | 5, 40 | mpd 13 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 ax-arch 7763 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 |
This theorem is referenced by: qbtwnz 10060 apbtwnz 10078 |
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