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Mirrors > Home > ILE Home > Th. List > suplubti | Unicode version |
Description: A supremum is the least upper bound. See also supclti 6972 and supubti 6973. (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 3990 | . . . . . . . 8 | |
3 | breq1 3990 | . . . . . . . . 9 | |
4 | 3 | rexbidv 2471 | . . . . . . . 8 |
5 | 2, 4 | imbi12d 233 | . . . . . . 7 |
6 | 5 | cbvralv 2696 | . . . . . 6 |
7 | 1, 6 | sylib 121 | . . . . 5 |
8 | 7 | a1i 9 | . . . 4 |
9 | 8 | ss2rabi 3229 | . . 3 |
10 | supmoti.ti | . . . . 5 | |
11 | supclti.2 | . . . . 5 | |
12 | 10, 11 | supval2ti 6969 | . . . 4 |
13 | 10, 11 | supeuti 6968 | . . . . 5 |
14 | riotacl2 5820 | . . . . 5 | |
15 | 13, 14 | syl 14 | . . . 4 |
16 | 12, 15 | eqeltrd 2247 | . . 3 |
17 | 9, 16 | sselid 3145 | . 2 |
18 | breq2 3991 | . . . . . 6 | |
19 | 18 | imbi1d 230 | . . . . 5 |
20 | 19 | ralbidv 2470 | . . . 4 |
21 | 20 | elrab 2886 | . . 3 |
22 | 21 | simprbi 273 | . 2 |
23 | breq1 3990 | . . . . 5 | |
24 | breq1 3990 | . . . . . 6 | |
25 | 24 | rexbidv 2471 | . . . . 5 |
26 | 23, 25 | imbi12d 233 | . . . 4 |
27 | 26 | rspccv 2831 | . . 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 1348 wcel 2141 wral 2448 wrex 2449 wreu 2450 crab 2452 class class class wbr 3987 crio 5806 csup 6956 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-iota 5158 df-riota 5807 df-sup 6958 |
This theorem is referenced by: suplub2ti 6975 supisoti 6984 infglbti 6999 sup3exmid 8862 maxleast 11166 |
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