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Mirrors > Home > ILE Home > Th. List > elprnql | Unicode 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prssnql 6669 | . 2 | |
2 | 1 | sselda 2999 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wcel 1433 cop 3401 cnq 6470 cnp 6481 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-qs 6135 df-ni 6494 df-nqqs 6538 df-inp 6656 |
This theorem is referenced by: prubl 6676 prnmaxl 6678 prarloclemlt 6683 prarloclemlo 6684 prarloclem5 6690 genpdf 6698 genipv 6699 genpelvl 6702 genpml 6707 genprndl 6711 genpassl 6714 addnqprllem 6717 addnqprl 6719 addlocprlemeqgt 6722 addlocprlemgt 6724 addlocprlem 6725 nqprl 6741 prmuloc 6756 mulnqprl 6758 addcomprg 6768 mulcomprg 6770 distrlem1prl 6772 distrlem4prl 6774 1idprl 6780 ltsopr 6786 ltexprlemm 6790 ltexprlemopl 6791 ltexprlemopu 6793 ltexprlemupu 6794 ltexprlemdisj 6796 ltexprlemloc 6797 ltexprlemfl 6799 ltexprlemrl 6800 ltexprlemfu 6801 ltexprlemru 6802 addcanprleml 6804 addcanprlemu 6805 recexprlemloc 6821 recexprlem1ssl 6823 recexprlem1ssu 6824 recexprlemss1l 6825 aptiprleml 6829 aptiprlemu 6830 caucvgprprlemopl 6887 |
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