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Mirrors > Home > ILE Home > Th. List > elprnqu | Unicode 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prssnqu 7412 | . 2 | |
2 | 1 | sselda 3137 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2135 cop 3573 cnq 7212 cnp 7223 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-qs 6498 df-ni 7236 df-nqqs 7280 df-inp 7398 |
This theorem is referenced by: prltlu 7419 prnminu 7421 genpdf 7440 genipv 7441 genpelvu 7445 genpmu 7450 genprndu 7454 genpassu 7457 addnqprulem 7460 addnqpru 7462 addlocprlemeqgt 7464 nqpru 7484 prmuloc 7498 mulnqpru 7501 addcomprg 7510 mulcomprg 7512 distrlem1pru 7515 distrlem4pru 7517 1idpru 7523 ltsopr 7528 ltaddpr 7529 ltexprlemm 7532 ltexprlemopl 7533 ltexprlemlol 7534 ltexprlemopu 7535 ltexprlemdisj 7538 ltexprlemloc 7539 ltexprlemfu 7543 ltexprlemru 7544 addcanprlemu 7547 prplnqu 7552 recexprlemloc 7563 recexprlemss1u 7568 aptiprlemu 7572 |
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