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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 6398 | . 2 inf | |
2 | df-sup 6397 | . . 3 | |
3 | ssel2 2994 | . . . . . . . . . 10 | |
4 | lenlt 7187 | . . . . . . . . . . 11 | |
5 | vex 2604 | . . . . . . . . . . . . 13 | |
6 | vex 2604 | . . . . . . . . . . . . 13 | |
7 | 5, 6 | brcnv 4536 | . . . . . . . . . . . 12 |
8 | 7 | notbii 626 | . . . . . . . . . . 11 |
9 | 4, 8 | syl6rbbr 197 | . . . . . . . . . 10 |
10 | 3, 9 | sylan2 280 | . . . . . . . . 9 |
11 | 10 | ancoms 264 | . . . . . . . 8 |
12 | 11 | an32s 532 | . . . . . . 7 |
13 | 12 | ralbidva 2364 | . . . . . 6 |
14 | 6, 5 | brcnv 4536 | . . . . . . . . 9 |
15 | vex 2604 | . . . . . . . . . . 11 | |
16 | 6, 15 | brcnv 4536 | . . . . . . . . . 10 |
17 | 16 | rexbii 2373 | . . . . . . . . 9 |
18 | 14, 17 | imbi12i 237 | . . . . . . . 8 |
19 | 18 | ralbii 2372 | . . . . . . 7 |
20 | 19 | a1i 9 | . . . . . 6 |
21 | 13, 20 | anbi12d 456 | . . . . 5 |
22 | 21 | rabbidva 2592 | . . . 4 |
23 | 22 | unieqd 3612 | . . 3 |
24 | 2, 23 | syl5eq 2125 | . 2 |
25 | 1, 24 | syl5eq 2125 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wral 2348 wrex 2349 crab 2352 wss 2973 cuni 3601 class class class wbr 3785 ccnv 4362 csup 6395 infcinf 6396 cr 6980 clt 7153 cle 7154 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-sup 6397 df-inf 6398 df-xr 7157 df-le 7159 |
This theorem is referenced by: (None) |
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