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Mirrors > Home > ILE Home > Th. List > prloc | Unicode version |
Description: A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.) |
Ref | Expression |
---|---|
prloc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elinp 7472 |
. . . 4
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2 | simpr3 1005 |
. . . 4
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3 | 1, 2 | sylbi 121 |
. . 3
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4 | 3 | adantr 276 |
. 2
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5 | simpr 110 |
. 2
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6 | ltrelnq 7363 |
. . . . . . 7
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7 | 6 | brel 4678 |
. . . . . 6
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8 | 7 | simpld 112 |
. . . . 5
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9 | 8 | adantl 277 |
. . . 4
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10 | simpr 110 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 10 | breq1d 4013 |
. . . . . 6
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12 | 10 | eleq1d 2246 |
. . . . . . 7
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13 | 12 | orbi1d 791 |
. . . . . 6
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14 | 11, 13 | imbi12d 234 |
. . . . 5
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15 | 14 | ralbidv 2477 |
. . . 4
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16 | 9, 15 | rspcdv 2844 |
. . 3
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17 | 7 | simprd 114 |
. . . . 5
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18 | 17 | adantl 277 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | simpr 110 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | 19 | breq2d 4015 |
. . . . 5
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21 | 19 | eleq1d 2246 |
. . . . . 6
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22 | 21 | orbi2d 790 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 20, 22 | imbi12d 234 |
. . . 4
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24 | 18, 23 | rspcdv 2844 |
. . 3
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25 | 16, 24 | syld 45 |
. 2
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26 | 4, 5, 25 | mp2d 47 |
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: prarloclem3step 7494 addnqprlemfl 7557 addnqprlemfu 7558 mullocprlem 7568 mulnqprlemfl 7573 mulnqprlemfu 7574 ltsopr 7594 ltexprlemloc 7605 addcanprleml 7612 addcanprlemu 7613 recexprlemloc 7629 cauappcvgprlemladdru 7654 cauappcvgprlemladdrl 7655 caucvgprlemladdrl 7676 |
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