Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6413 | . . . 4 | |
2 | opelxpi 4376 | . . . 4 | |
3 | 1, 2 | mpan2 401 | . . 3 |
4 | enqex 6458 | . . . 4 | |
5 | 4 | ecelqsi 6160 | . . 3 |
6 | 3, 5 | syl 14 | . 2 |
7 | df-nqqs 6446 | . 2 | |
8 | 6, 7 | syl6eleqr 2131 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1393 cop 3378 cxp 4343 c1o 5994 cec 6104 cqs 6105 cnpi 6370 ceq 6377 cnq 6378 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-suc 4108 df-iom 4314 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-1o 6001 df-ec 6108 df-qs 6112 df-ni 6402 df-enq 6445 df-nqqs 6446 |
This theorem is referenced by: recnnpr 6646 nnprlu 6651 archrecnq 6761 archrecpr 6762 caucvgprlemnkj 6764 caucvgprlemnbj 6765 caucvgprlemm 6766 caucvgprlemopl 6767 caucvgprlemlol 6768 caucvgprlemloc 6773 caucvgprlemladdfu 6775 caucvgprlemladdrl 6776 caucvgprprlemloccalc 6782 caucvgprprlemnkltj 6787 caucvgprprlemnkeqj 6788 caucvgprprlemnjltk 6789 caucvgprprlemml 6792 caucvgprprlemopl 6795 caucvgprprlemlol 6796 caucvgprprlemloc 6801 caucvgprprlemexb 6805 caucvgprprlem1 6807 caucvgprprlem2 6808 pitonnlem2 6923 ltrennb 6930 recidpipr 6932 |
Copyright terms: Public domain | W3C validator |