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Mirrors > Home > ILE Home > Th. List > supubti | Unicode version |
Description: A supremum is an upper
bound. See also supclti 6878 and suplubti 6880.
This proof demonstrates how to expand an iota-based definition (df-iota 5083) using riotacl2 5736. (Contributed by Jim Kingdon, 24-Nov-2021.) |
Ref | Expression |
---|---|
supmoti.ti | |
supclti.2 |
Ref | Expression |
---|---|
supubti |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . . 5 | |
2 | 1 | a1i 9 | . . . 4 |
3 | 2 | ss2rabi 3174 | . . 3 |
4 | supmoti.ti | . . . . 5 | |
5 | supclti.2 | . . . . 5 | |
6 | 4, 5 | supval2ti 6875 | . . . 4 |
7 | 4, 5 | supeuti 6874 | . . . . 5 |
8 | riotacl2 5736 | . . . . 5 | |
9 | 7, 8 | syl 14 | . . . 4 |
10 | 6, 9 | eqeltrd 2214 | . . 3 |
11 | 3, 10 | sseldi 3090 | . 2 |
12 | breq2 3928 | . . . . . . 7 | |
13 | 12 | notbid 656 | . . . . . 6 |
14 | 13 | cbvralv 2652 | . . . . 5 |
15 | breq1 3927 | . . . . . . 7 | |
16 | 15 | notbid 656 | . . . . . 6 |
17 | 16 | ralbidv 2435 | . . . . 5 |
18 | 14, 17 | syl5bb 191 | . . . 4 |
19 | 18 | elrab 2835 | . . 3 |
20 | 19 | simprbi 273 | . 2 |
21 | breq2 3928 | . . . 4 | |
22 | 21 | notbid 656 | . . 3 |
23 | 22 | rspccv 2781 | . 2 |
24 | 11, 20, 23 | 3syl 17 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2414 wrex 2415 wreu 2416 crab 2418 class class class wbr 3924 crio 5722 csup 6862 |
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 2119 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-iota 5083 df-riota 5723 df-sup 6864 |
This theorem is referenced by: suplub2ti 6881 supisoti 6890 inflbti 6904 suprubex 8702 zsupcl 11629 dvdslegcd 11642 |
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