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Mirrors > Home > ILE Home > Th. List > suprleubex | Unicode version |
Description: The supremum of a nonempty bounded set of reals is less than or equal to an upper bound. (Contributed by NM, 18-Mar-2005.) (Revised by Mario Carneiro, 6-Sep-2014.) |
Ref | Expression |
---|---|
suprubex.ex | |
suprubex.ss | |
suprlubex.b |
Ref | Expression |
---|---|
suprleubex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lttri3 7957 | . . . . . . . 8 | |
2 | 1 | adantl 275 | . . . . . . 7 |
3 | suprubex.ex | . . . . . . 7 | |
4 | 2, 3 | supclti 6942 | . . . . . 6 |
5 | suprlubex.b | . . . . . 6 | |
6 | 4, 5 | lenltd 7993 | . . . . 5 |
7 | suprubex.ss | . . . . . 6 | |
8 | 3, 7, 5 | suprnubex 8824 | . . . . 5 |
9 | 6, 8 | bitrd 187 | . . . 4 |
10 | breq2 3969 | . . . . . 6 | |
11 | 10 | notbid 657 | . . . . 5 |
12 | 11 | cbvralv 2680 | . . . 4 |
13 | 9, 12 | bitr4di 197 | . . 3 |
14 | 7 | sselda 3128 | . . . . 5 |
15 | 5 | adantr 274 | . . . . 5 |
16 | 14, 15 | lenltd 7993 | . . . 4 |
17 | 16 | ralbidva 2453 | . . 3 |
18 | 13, 17 | bitr4d 190 | . 2 |
19 | breq1 3968 | . . 3 | |
20 | 19 | cbvralv 2680 | . 2 |
21 | 18, 20 | bitrdi 195 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wcel 2128 wral 2435 wrex 2436 wss 3102 class class class wbr 3965 csup 6926 cr 7731 clt 7912 cle 7913 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-po 4256 df-iso 4257 df-xp 4592 df-cnv 4594 df-iota 5135 df-riota 5780 df-sup 6928 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 |
This theorem is referenced by: suprzclex 9262 suplociccex 13003 |
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