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Mirrors > Home > ILE Home > Th. List > prubl | Unicode version |
Description: A positive fraction not in a lower cut is an upper bound. (Contributed by Jim Kingdon, 29-Sep-2019.) |
Ref | Expression |
---|---|
prubl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2233 | . . . . . . 7 | |
2 | 1 | biimpcd 158 | . . . . . 6 |
3 | 2 | adantl 275 | . . . . 5 |
4 | prcdnql 7435 | . . . . 5 | |
5 | 3, 4 | jaod 712 | . . . 4 |
6 | 5 | con3d 626 | . . 3 |
7 | 6 | adantr 274 | . 2 |
8 | elprnql 7432 | . . 3 | |
9 | nqtric 7350 | . . 3 | |
10 | 8, 9 | sylan 281 | . 2 |
11 | 7, 10 | sylibrd 168 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 wceq 1348 wcel 2141 cop 3584 class class class wbr 3987 cnq 7231 cltq 7236 cnp 7242 |
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-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 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-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-eprel 4272 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-oadd 6397 df-omul 6398 df-er 6510 df-ec 6512 df-qs 6516 df-ni 7255 df-mi 7257 df-lti 7258 df-enq 7298 df-nqqs 7299 df-ltnqqs 7304 df-inp 7417 |
This theorem is referenced by: prltlu 7438 |
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