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Mirrors > Home > ILE Home > Th. List > nnnq | Unicode version |
Description: The canonical embedding of positive integers into positive fractions. (Contributed by Jim Kingdon, 26-Apr-2020.) |
Ref | Expression |
---|---|
nnnq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pi 7247 | . . . 4 | |
2 | opelxpi 4630 | . . . 4 | |
3 | 1, 2 | mpan2 422 | . . 3 |
4 | enqex 7292 | . . . 4 | |
5 | 4 | ecelqsi 6546 | . . 3 |
6 | 3, 5 | syl 14 | . 2 |
7 | df-nqqs 7280 | . 2 | |
8 | 6, 7 | eleqtrrdi 2258 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2135 cop 3573 cxp 4596 c1o 6368 cec 6490 cqs 6491 cnpi 7204 ceq 7211 cnq 7212 |
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-sep 4094 ax-nul 4102 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-ne 2335 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-suc 4343 df-iom 4562 df-xp 4604 df-cnv 4606 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-1o 6375 df-ec 6494 df-qs 6498 df-ni 7236 df-enq 7279 df-nqqs 7280 |
This theorem is referenced by: recnnpr 7480 nnprlu 7485 archrecnq 7595 archrecpr 7596 caucvgprlemnkj 7598 caucvgprlemnbj 7599 caucvgprlemm 7600 caucvgprlemopl 7601 caucvgprlemlol 7602 caucvgprlemloc 7607 caucvgprlemladdfu 7609 caucvgprlemladdrl 7610 caucvgprprlemloccalc 7616 caucvgprprlemnkltj 7621 caucvgprprlemnkeqj 7622 caucvgprprlemnjltk 7623 caucvgprprlemml 7626 caucvgprprlemopl 7629 caucvgprprlemlol 7630 caucvgprprlemloc 7635 caucvgprprlemexb 7639 caucvgprprlem1 7641 caucvgprprlem2 7642 pitonnlem2 7779 ltrennb 7786 recidpipr 7788 |
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