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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 7197 |
. . . . . 6
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2 | 1 | brel 4599 |
. . . . 5
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3 | 2 | simprd 113 |
. . . 4
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4 | 3 | adantl 275 |
. . 3
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5 | breq2 3941 |
. . . . . . 7
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6 | eleq1 2203 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 5, 6 | imbi12d 233 |
. . . . . 6
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8 | 7 | imbi2d 229 |
. . . . 5
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9 | 1 | brel 4599 |
. . . . . . . 8
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10 | an42 577 |
. . . . . . . . 9
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11 | breq1 3940 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | eleq1 2203 |
. . . . . . . . . . . . . . . 16
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 11, 12 | anbi12d 465 |
. . . . . . . . . . . . . . 15
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14 | 13 | rspcev 2793 |
. . . . . . . . . . . . . 14
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15 | elinp 7306 |
. . . . . . . . . . . . . . . 16
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16 | simpr1r 1040 |
. . . . . . . . . . . . . . . 16
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17 | 15, 16 | sylbi 120 |
. . . . . . . . . . . . . . 15
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18 | 17 | r19.21bi 2523 |
. . . . . . . . . . . . . 14
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19 | 14, 18 | syl5ibrcom 156 |
. . . . . . . . . . . . 13
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20 | 19 | 3impb 1178 |
. . . . . . . . . . . 12
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21 | 20 | 3com12 1186 |
. . . . . . . . . . 11
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22 | 21 | 3expib 1185 |
. . . . . . . . . 10
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23 | 22 | impd 252 |
. . . . . . . . 9
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24 | 10, 23 | syl5bi 151 |
. . . . . . . 8
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25 | 9, 24 | mpand 426 |
. . . . . . 7
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26 | 25 | com12 30 |
. . . . . 6
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27 | 26 | ancoms 266 |
. . . . 5
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28 | 8, 27 | vtoclg 2749 |
. . . 4
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29 | 28 | impd 252 |
. . 3
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30 | 4, 29 | mpcom 36 |
. 2
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31 | 30 | ex 114 |
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 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-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 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-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 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-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-qs 6443 df-ni 7136 df-nqqs 7180 df-ltnqqs 7185 df-inp 7298 |
This theorem is referenced by: prarloc 7335 prarloc2 7336 addnqprulem 7360 nqpru 7384 prmuloc2 7399 mulnqpru 7401 distrlem4pru 7417 1idpru 7423 ltexprlemm 7432 ltexprlemupu 7436 ltexprlemrl 7442 ltexprlemfu 7443 ltexprlemru 7444 aptiprlemu 7472 suplocexprlemdisj 7552 suplocexprlemub 7555 |
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