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Mirrors > Home > ILE Home > Th. List > suplubti | Unicode version |
Description: A supremum is the least upper bound. See also supclti 6885 and supubti 6886. (Contributed by Jim Kingdon, 24-Nov-2021.) |
Ref | Expression |
---|---|
supmoti.ti | |
supclti.2 |
Ref | Expression |
---|---|
suplubti |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . . 6 | |
2 | breq1 3932 | . . . . . . . 8 | |
3 | breq1 3932 | . . . . . . . . 9 | |
4 | 3 | rexbidv 2438 | . . . . . . . 8 |
5 | 2, 4 | imbi12d 233 | . . . . . . 7 |
6 | 5 | cbvralv 2654 | . . . . . 6 |
7 | 1, 6 | sylib 121 | . . . . 5 |
8 | 7 | a1i 9 | . . . 4 |
9 | 8 | ss2rabi 3179 | . . 3 |
10 | supmoti.ti | . . . . 5 | |
11 | supclti.2 | . . . . 5 | |
12 | 10, 11 | supval2ti 6882 | . . . 4 |
13 | 10, 11 | supeuti 6881 | . . . . 5 |
14 | riotacl2 5743 | . . . . 5 | |
15 | 13, 14 | syl 14 | . . . 4 |
16 | 12, 15 | eqeltrd 2216 | . . 3 |
17 | 9, 16 | sseldi 3095 | . 2 |
18 | breq2 3933 | . . . . . 6 | |
19 | 18 | imbi1d 230 | . . . . 5 |
20 | 19 | ralbidv 2437 | . . . 4 |
21 | 20 | elrab 2840 | . . 3 |
22 | 21 | simprbi 273 | . 2 |
23 | breq1 3932 | . . . . 5 | |
24 | breq1 3932 | . . . . . 6 | |
25 | 24 | rexbidv 2438 | . . . . 5 |
26 | 23, 25 | imbi12d 233 | . . . 4 |
27 | 26 | rspccv 2786 | . . 3 |
28 | 27 | impd 252 | . 2 |
29 | 17, 22, 28 | 3syl 17 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wrex 2417 wreu 2418 crab 2420 class class class wbr 3929 crio 5729 csup 6869 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 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-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-riota 5730 df-sup 6871 |
This theorem is referenced by: suplub2ti 6888 supisoti 6897 infglbti 6912 sup3exmid 8715 maxleast 10985 |
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