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Mirrors > Home > ILE Home > Th. List > dfinfre | Unicode version |
Description: The infimum of a set of reals . (Contributed by NM, 9-Oct-2005.) (Revised by AV, 4-Sep-2020.) |
Ref | Expression |
---|---|
dfinfre | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 6872 | . 2 inf | |
2 | df-sup 6871 | . . 3 | |
3 | ssel2 3092 | . . . . . . . . . 10 | |
4 | lenlt 7840 | . . . . . . . . . . 11 | |
5 | vex 2689 | . . . . . . . . . . . . 13 | |
6 | vex 2689 | . . . . . . . . . . . . 13 | |
7 | 5, 6 | brcnv 4722 | . . . . . . . . . . . 12 |
8 | 7 | notbii 657 | . . . . . . . . . . 11 |
9 | 4, 8 | syl6rbbr 198 | . . . . . . . . . 10 |
10 | 3, 9 | sylan2 284 | . . . . . . . . 9 |
11 | 10 | ancoms 266 | . . . . . . . 8 |
12 | 11 | an32s 557 | . . . . . . 7 |
13 | 12 | ralbidva 2433 | . . . . . 6 |
14 | 6, 5 | brcnv 4722 | . . . . . . . . 9 |
15 | vex 2689 | . . . . . . . . . . 11 | |
16 | 6, 15 | brcnv 4722 | . . . . . . . . . 10 |
17 | 16 | rexbii 2442 | . . . . . . . . 9 |
18 | 14, 17 | imbi12i 238 | . . . . . . . 8 |
19 | 18 | ralbii 2441 | . . . . . . 7 |
20 | 19 | a1i 9 | . . . . . 6 |
21 | 13, 20 | anbi12d 464 | . . . . 5 |
22 | 21 | rabbidva 2674 | . . . 4 |
23 | 22 | unieqd 3747 | . . 3 |
24 | 2, 23 | syl5eq 2184 | . 2 |
25 | 1, 24 | syl5eq 2184 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wrex 2417 crab 2420 wss 3071 cuni 3736 class class class wbr 3929 ccnv 4538 csup 6869 infcinf 6870 cr 7619 clt 7800 cle 7801 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-sup 6871 df-inf 6872 df-xr 7804 df-le 7806 |
This theorem is referenced by: (None) |
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