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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 7255 | . 2 ⊢ (〈𝐿, 𝑈〉 ∈ P → 𝐿 ⊆ Q) | |
2 | 1 | sselda 3067 | 1 ⊢ ((〈𝐿, 𝑈〉 ∈ P ∧ 𝐵 ∈ 𝐿) → 𝐵 ∈ Q) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1465 〈cop 3500 Qcnq 7056 Pcnp 7067 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-qs 6403 df-ni 7080 df-nqqs 7124 df-inp 7242 |
This theorem is referenced by: prubl 7262 prnmaxl 7264 prarloclemlt 7269 prarloclemlo 7270 prarloclem5 7276 genpdf 7284 genipv 7285 genpelvl 7288 genpml 7293 genprndl 7297 genpassl 7300 addnqprllem 7303 addnqprl 7305 addlocprlemeqgt 7308 addlocprlemgt 7310 addlocprlem 7311 nqprl 7327 prmuloc 7342 mulnqprl 7344 addcomprg 7354 mulcomprg 7356 distrlem1prl 7358 distrlem4prl 7360 1idprl 7366 ltsopr 7372 ltexprlemm 7376 ltexprlemopl 7377 ltexprlemopu 7379 ltexprlemupu 7380 ltexprlemdisj 7382 ltexprlemloc 7383 ltexprlemfl 7385 ltexprlemrl 7386 ltexprlemfu 7387 ltexprlemru 7388 addcanprleml 7390 addcanprlemu 7391 recexprlemloc 7407 recexprlem1ssl 7409 recexprlem1ssu 7410 recexprlemss1l 7411 aptiprleml 7415 aptiprlemu 7416 caucvgprprlemopl 7473 suplocexprlemex 7498 |
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