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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 7363 |
. . . . . 6
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2 | 1 | brel 4678 |
. . . . 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 4007 |
. . . . . . 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 4678 |
. . . . . . . 8
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10 | an42 587 |
. . . . . . . . 9
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11 | breq1 4006 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | eleq1 2240 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 11, 12 | anbi12d 473 |
. . . . . . . . . . . . . . 15
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14 | 13 | rspcev 2841 |
. . . . . . . . . . . . . 14
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15 | elinp 7472 |
. . . . . . . . . . . . . . . 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 2797 |
. . . 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 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-iinf 4587 |
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 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-qs 6540 df-ni 7302 df-nqqs 7346 df-ltnqqs 7351 df-inp 7464 |
This theorem is referenced by: prarloc 7501 prarloc2 7502 addnqprulem 7526 nqpru 7550 prmuloc2 7565 mulnqpru 7567 distrlem4pru 7583 1idpru 7589 ltexprlemm 7598 ltexprlemupu 7602 ltexprlemrl 7608 ltexprlemfu 7609 ltexprlemru 7610 aptiprlemu 7638 suplocexprlemdisj 7718 suplocexprlemub 7721 |
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