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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 6941 | . 2 inf | |
2 | df-sup 6940 | . . 3 | |
3 | ssel2 3132 | . . . . . . . . . 10 | |
4 | vex 2724 | . . . . . . . . . . . . 13 | |
5 | vex 2724 | . . . . . . . . . . . . 13 | |
6 | 4, 5 | brcnv 4781 | . . . . . . . . . . . 12 |
7 | 6 | notbii 658 | . . . . . . . . . . 11 |
8 | lenlt 7965 | . . . . . . . . . . 11 | |
9 | 7, 8 | bitr4id 198 | . . . . . . . . . 10 |
10 | 3, 9 | sylan2 284 | . . . . . . . . 9 |
11 | 10 | ancoms 266 | . . . . . . . 8 |
12 | 11 | an32s 558 | . . . . . . 7 |
13 | 12 | ralbidva 2460 | . . . . . 6 |
14 | 5, 4 | brcnv 4781 | . . . . . . . . 9 |
15 | vex 2724 | . . . . . . . . . . 11 | |
16 | 5, 15 | brcnv 4781 | . . . . . . . . . 10 |
17 | 16 | rexbii 2471 | . . . . . . . . 9 |
18 | 14, 17 | imbi12i 238 | . . . . . . . 8 |
19 | 18 | ralbii 2470 | . . . . . . 7 |
20 | 19 | a1i 9 | . . . . . 6 |
21 | 13, 20 | anbi12d 465 | . . . . 5 |
22 | 21 | rabbidva 2709 | . . . 4 |
23 | 22 | unieqd 3794 | . . 3 |
24 | 2, 23 | syl5eq 2209 | . 2 |
25 | 1, 24 | syl5eq 2209 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 wrex 2443 crab 2446 wss 3111 cuni 3783 class class class wbr 3976 ccnv 4597 csup 6938 infcinf 6939 cr 7743 clt 7924 cle 7925 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 df-sup 6940 df-inf 6941 df-xr 7928 df-le 7930 |
This theorem is referenced by: (None) |
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