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Mirrors > Home > ILE Home > Th. List > prcunqu | Unicode version |
Description: An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.) |
Ref | Expression |
---|---|
prcunqu |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 7364 |
. . . . . 6
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2 | 1 | brel 4679 |
. . . . 5
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3 | 2 | simprd 114 |
. . . 4
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4 | 3 | adantl 277 |
. . 3
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5 | breq2 4008 |
. . . . . . 7
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6 | eleq1 2240 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | imbi12d 234 |
. . . . . 6
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8 | 7 | imbi2d 230 |
. . . . 5
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9 | 1 | brel 4679 |
. . . . . . . 8
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10 | an42 587 |
. . . . . . . . 9
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11 | breq1 4007 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | eleq1 2240 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 11, 12 | anbi12d 473 |
. . . . . . . . . . . . . . 15
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14 | 13 | rspcev 2842 |
. . . . . . . . . . . . . 14
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15 | elinp 7473 |
. . . . . . . . . . . . . . . 16
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16 | simpr1r 1055 |
. . . . . . . . . . . . . . . 16
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17 | 15, 16 | sylbi 121 |
. . . . . . . . . . . . . . 15
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18 | 17 | r19.21bi 2565 |
. . . . . . . . . . . . . 14
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19 | 14, 18 | syl5ibrcom 157 |
. . . . . . . . . . . . 13
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20 | 19 | 3impb 1199 |
. . . . . . . . . . . 12
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21 | 20 | 3com12 1207 |
. . . . . . . . . . 11
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22 | 21 | 3expib 1206 |
. . . . . . . . . 10
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23 | 22 | impd 254 |
. . . . . . . . 9
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24 | 10, 23 | biimtrid 152 |
. . . . . . . 8
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25 | 9, 24 | mpand 429 |
. . . . . . 7
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26 | 25 | com12 30 |
. . . . . 6
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27 | 26 | ancoms 268 |
. . . . 5
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28 | 8, 27 | vtoclg 2798 |
. . . 4
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29 | 28 | impd 254 |
. . 3
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30 | 4, 29 | mpcom 36 |
. 2
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31 | 30 | ex 115 |
1
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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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-iinf 4588 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-qs 6541 df-ni 7303 df-nqqs 7347 df-ltnqqs 7352 df-inp 7465 |
This theorem is referenced by: prarloc 7502 prarloc2 7503 addnqprulem 7527 nqpru 7551 prmuloc2 7566 mulnqpru 7568 distrlem4pru 7584 1idpru 7590 ltexprlemm 7599 ltexprlemupu 7603 ltexprlemrl 7609 ltexprlemfu 7610 ltexprlemru 7611 aptiprlemu 7639 suplocexprlemdisj 7719 suplocexprlemub 7722 |
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