![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > elprnql | GIF version |
Description: An element of a positive real's lower cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.) |
Ref | Expression |
---|---|
elprnql | ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝐵 ∈ 𝐿) → 𝐵 ∈ Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prssnql 7509 | . 2 ⊢ (〈𝐿, 𝑈〉 ∈ P → 𝐿 ⊆ Q) | |
2 | 1 | sselda 3170 | 1 ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝐵 ∈ 𝐿) → 𝐵 ∈ Q) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2160 〈cop 3610 Qcnq 7310 Pcnp 7321 |
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 6566 df-ni 7334 df-nqqs 7378 df-inp 7496 |
This theorem is referenced by: prubl 7516 prnmaxl 7518 prarloclemlt 7523 prarloclemlo 7524 prarloclem5 7530 genpdf 7538 genipv 7539 genpelvl 7542 genpml 7547 genprndl 7551 genpassl 7554 addnqprllem 7557 addnqprl 7559 addlocprlemeqgt 7562 addlocprlemgt 7564 addlocprlem 7565 nqprl 7581 prmuloc 7596 mulnqprl 7598 addcomprg 7608 mulcomprg 7610 distrlem1prl 7612 distrlem4prl 7614 1idprl 7620 ltsopr 7626 ltexprlemm 7630 ltexprlemopl 7631 ltexprlemopu 7633 ltexprlemupu 7634 ltexprlemdisj 7636 ltexprlemloc 7637 ltexprlemfl 7639 ltexprlemrl 7640 ltexprlemfu 7641 ltexprlemru 7642 addcanprleml 7644 addcanprlemu 7645 recexprlemloc 7661 recexprlem1ssl 7663 recexprlem1ssu 7664 recexprlemss1l 7665 aptiprleml 7669 aptiprlemu 7670 caucvgprprlemopl 7727 suplocexprlemex 7752 |
Copyright terms: Public domain | W3C validator |