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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 9403 |
. . . 4
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3 | 1, 2 | syl 14 |
. . 3
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4 | reeanv 2660 |
. . 3
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5 | 3, 4 | sylibr 134 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | simplrl 535 |
. . . . . 6
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7 | 6 | zred 9406 |
. . . . . . 7
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8 | 1 | ad2antrr 488 |
. . . . . . 7
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9 | simprl 529 |
. . . . . . 7
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10 | 7, 8, 9 | ltled 8107 |
. . . . . 6
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11 | simprr 531 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 6 | zcnd 9407 |
. . . . . . . 8
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13 | simplrr 536 |
. . . . . . . . 9
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14 | 13 | zcnd 9407 |
. . . . . . . 8
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15 | 12, 14 | pncan3d 8302 |
. . . . . . 7
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16 | 11, 15 | breqtrrd 4046 |
. . . . . 6
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17 | breq1 4021 |
. . . . . . . 8
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18 | oveq1 5904 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 18 | breq2d 4030 |
. . . . . . . 8
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20 | 17, 19 | anbi12d 473 |
. . . . . . 7
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21 | 20 | rspcev 2856 |
. . . . . 6
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22 | 6, 10, 16, 21 | syl12anc 1247 |
. . . . 5
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23 | 13 | zred 9406 |
. . . . . . . 8
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24 | 7, 8, 23, 9, 11 | lttrd 8114 |
. . . . . . 7
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25 | znnsub 9335 |
. . . . . . . 8
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26 | 25 | ad2antlr 489 |
. . . . . . 7
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27 | 24, 26 | mpbid 147 |
. . . . . 6
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28 | exbtwnzlemex.tri |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | 28 | ralrimiva 2563 |
. . . . . . . . 9
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30 | breq1 4021 |
. . . . . . . . . . 11
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31 | breq2 4022 |
. . . . . . . . . . 11
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32 | 30, 31 | orbi12d 794 |
. . . . . . . . . 10
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33 | 32 | cbvralv 2718 |
. . . . . . . . 9
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34 | 29, 33 | sylib 122 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 34 | ad2antrr 488 |
. . . . . . 7
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36 | 35 | r19.21bi 2578 |
. . . . . 6
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37 | 27, 8, 36 | exbtwnzlemshrink 10281 |
. . . . 5
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38 | 22, 37 | mpdan 421 |
. . . 4
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39 | 38 | ex 115 |
. . 3
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40 | 39 | rexlimdvva 2615 |
. 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 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-ltadd 7958 ax-arch 7961 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-inn 8951 df-n0 9208 df-z 9285 |
This theorem is referenced by: qbtwnz 10284 apbtwnz 10307 |
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