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Mirrors > Home > ILE Home > Th. List > suplubti | Unicode version |
Description: A supremum is the least upper bound. See also supclti 6411 and supubti 6412. (Contributed by Jim Kingdon, 24-Nov-2021.) |
Ref | Expression |
---|---|
supmoti.ti | |
supclti.2 |
Ref | Expression |
---|---|
suplubti |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 108 | . . . . . 6 | |
2 | breq1 3788 | . . . . . . . 8 | |
3 | breq1 3788 | . . . . . . . . 9 | |
4 | 3 | rexbidv 2369 | . . . . . . . 8 |
5 | 2, 4 | imbi12d 232 | . . . . . . 7 |
6 | 5 | cbvralv 2577 | . . . . . 6 |
7 | 1, 6 | sylib 120 | . . . . 5 |
8 | 7 | a1i 9 | . . . 4 |
9 | 8 | ss2rabi 3076 | . . 3 |
10 | supmoti.ti | . . . . 5 | |
11 | supclti.2 | . . . . 5 | |
12 | 10, 11 | supval2ti 6408 | . . . 4 |
13 | 10, 11 | supeuti 6407 | . . . . 5 |
14 | riotacl2 5501 | . . . . 5 | |
15 | 13, 14 | syl 14 | . . . 4 |
16 | 12, 15 | eqeltrd 2155 | . . 3 |
17 | 9, 16 | sseldi 2997 | . 2 |
18 | breq2 3789 | . . . . . 6 | |
19 | 18 | imbi1d 229 | . . . . 5 |
20 | 19 | ralbidv 2368 | . . . 4 |
21 | 20 | elrab 2749 | . . 3 |
22 | 21 | simprbi 269 | . 2 |
23 | breq1 3788 | . . . . 5 | |
24 | breq1 3788 | . . . . . 6 | |
25 | 24 | rexbidv 2369 | . . . . 5 |
26 | 23, 25 | imbi12d 232 | . . . 4 |
27 | 26 | rspccv 2698 | . . 3 |
28 | 27 | impd 251 | . 2 |
29 | 17, 22, 28 | 3syl 17 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wral 2348 wrex 2349 wreu 2350 crab 2352 class class class wbr 3785 crio 5487 csup 6395 |
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-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 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-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-riota 5488 df-sup 6397 |
This theorem is referenced by: suplub2ti 6414 supisoti 6423 infglbti 6438 maxleast 10099 |
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