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Mirrors > Home > ILE Home > Th. List > lbreu | Unicode version |
Description: If a set of reals contains a lower bound, it contains a unique lower bound. (Contributed by NM, 9-Oct-2005.) |
Ref | Expression |
---|---|
lbreu |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4022 |
. . . . . . . . 9
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2 | 1 | rspcv 2852 |
. . . . . . . 8
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3 | breq2 4022 |
. . . . . . . . 9
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4 | 3 | rspcv 2852 |
. . . . . . . 8
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5 | 2, 4 | im2anan9r 599 |
. . . . . . 7
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6 | ssel 3164 |
. . . . . . . . . . . 12
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7 | ssel 3164 |
. . . . . . . . . . . 12
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8 | 6, 7 | anim12d 335 |
. . . . . . . . . . 11
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9 | 8 | impcom 125 |
. . . . . . . . . 10
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10 | letri3 8067 |
. . . . . . . . . 10
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11 | 9, 10 | syl 14 |
. . . . . . . . 9
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12 | 11 | exbiri 382 |
. . . . . . . 8
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13 | 12 | com23 78 |
. . . . . . 7
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14 | 5, 13 | syld 45 |
. . . . . 6
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15 | 14 | com3r 79 |
. . . . 5
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16 | 15 | ralrimivv 2571 |
. . . 4
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17 | 16 | anim2i 342 |
. . 3
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18 | 17 | ancoms 268 |
. 2
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19 | breq1 4021 |
. . . 4
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20 | 19 | ralbidv 2490 |
. . 3
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21 | 20 | reu4 2946 |
. 2
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22 | 18, 21 | sylibr 134 |
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 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 7931 ax-resscn 7932 ax-pre-ltirr 7952 ax-pre-apti 7955 |
This theorem depends on definitions: df-bi 117 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-rmo 2476 df-rab 2477 df-v 2754 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-br 4019 df-opab 4080 df-xp 4650 df-cnv 4652 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 |
This theorem is referenced by: lbcl 8932 lble 8933 |
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