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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 3899 |
. . . . . . . . 9
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2 | 1 | rspcv 2756 |
. . . . . . . 8
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3 | breq2 3899 |
. . . . . . . . 9
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4 | 3 | rspcv 2756 |
. . . . . . . 8
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5 | 2, 4 | im2anan9r 571 |
. . . . . . 7
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6 | ssel 3057 |
. . . . . . . . . . . 12
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7 | ssel 3057 |
. . . . . . . . . . . 12
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8 | 6, 7 | anim12d 331 |
. . . . . . . . . . 11
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9 | 8 | impcom 124 |
. . . . . . . . . 10
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10 | letri3 7768 |
. . . . . . . . . 10
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11 | 9, 10 | syl 14 |
. . . . . . . . 9
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12 | 11 | exbiri 377 |
. . . . . . . 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 2487 |
. . . 4
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17 | 16 | anim2i 337 |
. . 3
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18 | 17 | ancoms 266 |
. 2
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19 | breq1 3898 |
. . . 4
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20 | 19 | ralbidv 2411 |
. . 3
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21 | 20 | reu4 2847 |
. 2
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22 | 18, 21 | sylibr 133 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-pre-ltirr 7657 ax-pre-apti 7660 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-reu 2397 df-rmo 2398 df-rab 2399 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-xp 4505 df-cnv 4507 df-pnf 7726 df-mnf 7727 df-xr 7728 df-ltxr 7729 df-le 7730 |
This theorem is referenced by: lbcl 8614 lble 8615 |
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