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Mirrors > Home > ILE Home > Th. List > suprubex | Unicode version |
Description: A member of a nonempty bounded set of reals is less than or equal to the set's upper bound. (Contributed by Jim Kingdon, 18-Jan-2022.) |
Ref | Expression |
---|---|
suprubex.ex | |
suprubex.ss | |
suprubex.b |
Ref | Expression |
---|---|
suprubex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suprubex.ss | . . 3 | |
2 | suprubex.b | . . 3 | |
3 | 1, 2 | sseldd 3068 | . 2 |
4 | lttri3 7812 | . . . 4 | |
5 | 4 | adantl 275 | . . 3 |
6 | suprubex.ex | . . 3 | |
7 | 5, 6 | supclti 6853 | . 2 |
8 | 5, 6 | supubti 6854 | . . 3 |
9 | 2, 8 | mpd 13 | . 2 |
10 | 3, 7, 9 | nltled 7851 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wcel 1465 wral 2393 wrex 2394 wss 3041 class class class wbr 3899 csup 6837 cr 7587 clt 7768 cle 7769 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltirr 7700 ax-pre-apti 7703 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-iota 5058 df-riota 5698 df-sup 6839 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 |
This theorem is referenced by: suprzclex 9117 |
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