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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 6840 | . 2 inf | |
2 | df-sup 6839 | . . 3 | |
3 | ssel2 3062 | . . . . . . . . . 10 | |
4 | lenlt 7808 | . . . . . . . . . . 11 | |
5 | vex 2663 | . . . . . . . . . . . . 13 | |
6 | vex 2663 | . . . . . . . . . . . . 13 | |
7 | 5, 6 | brcnv 4692 | . . . . . . . . . . . 12 |
8 | 7 | notbii 642 | . . . . . . . . . . 11 |
9 | 4, 8 | syl6rbbr 198 | . . . . . . . . . 10 |
10 | 3, 9 | sylan2 284 | . . . . . . . . 9 |
11 | 10 | ancoms 266 | . . . . . . . 8 |
12 | 11 | an32s 542 | . . . . . . 7 |
13 | 12 | ralbidva 2410 | . . . . . 6 |
14 | 6, 5 | brcnv 4692 | . . . . . . . . 9 |
15 | vex 2663 | . . . . . . . . . . 11 | |
16 | 6, 15 | brcnv 4692 | . . . . . . . . . 10 |
17 | 16 | rexbii 2419 | . . . . . . . . 9 |
18 | 14, 17 | imbi12i 238 | . . . . . . . 8 |
19 | 18 | ralbii 2418 | . . . . . . 7 |
20 | 19 | a1i 9 | . . . . . 6 |
21 | 13, 20 | anbi12d 464 | . . . . 5 |
22 | 21 | rabbidva 2648 | . . . 4 |
23 | 22 | unieqd 3717 | . . 3 |
24 | 2, 23 | syl5eq 2162 | . 2 |
25 | 1, 24 | syl5eq 2162 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wral 2393 wrex 2394 crab 2397 wss 3041 cuni 3706 class class class wbr 3899 ccnv 4508 csup 6837 infcinf 6838 cr 7587 clt 7768 cle 7769 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-sup 6839 df-inf 6840 df-xr 7772 df-le 7774 |
This theorem is referenced by: (None) |
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