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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1pi 7311 |
. . . 4
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2 | opelxpi 4657 |
. . . 4
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3 | 1, 2 | mpan2 425 |
. . 3
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4 | enqex 7356 |
. . . 4
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5 | 4 | ecelqsi 6586 |
. . 3
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6 | 3, 5 | syl 14 |
. 2
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7 | df-nqqs 7344 |
. 2
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8 | 6, 7 | eleqtrrdi 2271 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-iinf 4586 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-suc 4370 df-iom 4589 df-xp 4631 df-cnv 4633 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-1o 6414 df-ec 6534 df-qs 6538 df-ni 7300 df-enq 7343 df-nqqs 7344 |
This theorem is referenced by: recnnpr 7544 nnprlu 7549 archrecnq 7659 archrecpr 7660 caucvgprlemnkj 7662 caucvgprlemnbj 7663 caucvgprlemm 7664 caucvgprlemopl 7665 caucvgprlemlol 7666 caucvgprlemloc 7671 caucvgprlemladdfu 7673 caucvgprlemladdrl 7674 caucvgprprlemloccalc 7680 caucvgprprlemnkltj 7685 caucvgprprlemnkeqj 7686 caucvgprprlemnjltk 7687 caucvgprprlemml 7690 caucvgprprlemopl 7693 caucvgprprlemlol 7694 caucvgprprlemloc 7699 caucvgprprlemexb 7703 caucvgprprlem1 7705 caucvgprprlem2 7706 pitonnlem2 7843 ltrennb 7850 recidpipr 7852 |
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