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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 7312 | . 2 ⊢ (〈𝐿, 𝑈〉 ∈ P → 𝑈 ⊆ Q) | |
2 | 1 | sselda 3102 | 1 ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ Q) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1481 〈cop 3535 Qcnq 7112 Pcnp 7123 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-qs 6443 df-ni 7136 df-nqqs 7180 df-inp 7298 |
This theorem is referenced by: prltlu 7319 prnminu 7321 genpdf 7340 genipv 7341 genpelvu 7345 genpmu 7350 genprndu 7354 genpassu 7357 addnqprulem 7360 addnqpru 7362 addlocprlemeqgt 7364 nqpru 7384 prmuloc 7398 mulnqpru 7401 addcomprg 7410 mulcomprg 7412 distrlem1pru 7415 distrlem4pru 7417 1idpru 7423 ltsopr 7428 ltaddpr 7429 ltexprlemm 7432 ltexprlemopl 7433 ltexprlemlol 7434 ltexprlemopu 7435 ltexprlemdisj 7438 ltexprlemloc 7439 ltexprlemfu 7443 ltexprlemru 7444 addcanprlemu 7447 prplnqu 7452 recexprlemloc 7463 recexprlemss1u 7468 aptiprlemu 7472 |
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