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Mirrors > Home > ILE Home > Th. List > elprnqu | GIF version |
Description: An element of a positive real's upper cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.) |
Ref | Expression |
---|---|
elprnqu | ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ B ∈ 𝑈) → B ∈ Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prssnqu 6463 | . 2 ⊢ (〈𝐿, 𝑈〉 ∈ P → 𝑈 ⊆ Q) | |
2 | 1 | sselda 2939 | 1 ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ B ∈ 𝑈) → B ∈ Q) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 〈cop 3370 Qcnq 6264 Pcnp 6275 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-qs 6048 df-ni 6288 df-nqqs 6332 df-inp 6449 |
This theorem is referenced by: prltlu 6470 prnminu 6472 genpdf 6491 genipv 6492 genpelvu 6496 genpmu 6501 genprndu 6505 genpassu 6508 addnqprulem 6511 addnqpru 6513 addlocprlemeqgt 6515 prmuloc 6547 mulnqpru 6550 addcomprg 6554 mulcomprg 6556 distrlem1pru 6559 distrlem4pru 6561 1idpru 6567 ltsopr 6570 ltaddpr 6571 ltexprlemm 6574 ltexprlemopl 6575 ltexprlemlol 6576 ltexprlemopu 6577 ltexprlemdisj 6580 ltexprlemloc 6581 ltexprlemfu 6585 ltexprlemru 6586 addcanprlemu 6589 recexprlemloc 6603 recexprlemss1u 6608 aptiprlemu 6612 |
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