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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 7509 | . 2 ⊢ (〈𝐿, 𝑈〉 ∈ P → 𝑈 ⊆ Q) | |
2 | 1 | sselda 3170 | 1 ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝐵 ∈ 𝑈) → 𝐵 ∈ Q) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 〈cop 3610 Qcnq 7309 Pcnp 7320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-qs 6565 df-ni 7333 df-nqqs 7377 df-inp 7495 |
This theorem is referenced by: prltlu 7516 prnminu 7518 genpdf 7537 genipv 7538 genpelvu 7542 genpmu 7547 genprndu 7551 genpassu 7554 addnqprulem 7557 addnqpru 7559 addlocprlemeqgt 7561 nqpru 7581 prmuloc 7595 mulnqpru 7598 addcomprg 7607 mulcomprg 7609 distrlem1pru 7612 distrlem4pru 7614 1idpru 7620 ltsopr 7625 ltaddpr 7626 ltexprlemm 7629 ltexprlemopl 7630 ltexprlemlol 7631 ltexprlemopu 7632 ltexprlemdisj 7635 ltexprlemloc 7636 ltexprlemfu 7640 ltexprlemru 7641 addcanprlemu 7644 prplnqu 7649 recexprlemloc 7660 recexprlemss1u 7665 aptiprlemu 7669 |
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