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Mirrors > Home > ILE Home > Th. List > dfinfre | Unicode version |
Description: The infimum of a set of
reals ![]() |
Ref | Expression |
---|---|
dfinfre |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 6824 |
. 2
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2 | df-sup 6823 |
. . 3
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3 | ssel2 3058 |
. . . . . . . . . 10
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4 | lenlt 7763 |
. . . . . . . . . . 11
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5 | vex 2660 |
. . . . . . . . . . . . 13
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6 | vex 2660 |
. . . . . . . . . . . . 13
![]() ![]() ![]() ![]() | |
7 | 5, 6 | brcnv 4682 |
. . . . . . . . . . . 12
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8 | 7 | notbii 640 |
. . . . . . . . . . 11
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9 | 4, 8 | syl6rbbr 198 |
. . . . . . . . . 10
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10 | 3, 9 | sylan2 282 |
. . . . . . . . 9
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11 | 10 | ancoms 266 |
. . . . . . . 8
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12 | 11 | an32s 540 |
. . . . . . 7
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13 | 12 | ralbidva 2407 |
. . . . . 6
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14 | 6, 5 | brcnv 4682 |
. . . . . . . . 9
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15 | vex 2660 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() | |
16 | 6, 15 | brcnv 4682 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 16 | rexbii 2416 |
. . . . . . . . 9
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18 | 14, 17 | imbi12i 238 |
. . . . . . . 8
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19 | 18 | ralbii 2415 |
. . . . . . 7
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20 | 19 | a1i 9 |
. . . . . 6
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21 | 13, 20 | anbi12d 462 |
. . . . 5
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22 | 21 | rabbidva 2645 |
. . . 4
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23 | 22 | unieqd 3713 |
. . 3
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24 | 2, 23 | syl5eq 2159 |
. 2
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25 | 1, 24 | syl5eq 2159 |
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-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 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 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-ral 2395 df-rex 2396 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-sup 6823 df-inf 6824 df-xr 7728 df-le 7730 |
This theorem is referenced by: (None) |
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