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Mirrors > Home > ILE Home > Th. List > nqnq0 | Unicode version |
Description: A positive fraction is a nonnegative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.) |
Ref | Expression |
---|---|
nqnq0 | Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nqqs 7156 | . . . . 5 | |
2 | 1 | eleq2i 2206 | . . . 4 |
3 | vex 2689 | . . . . 5 | |
4 | 3 | elqs 6480 | . . . 4 |
5 | df-rex 2422 | . . . 4 | |
6 | 2, 4, 5 | 3bitri 205 | . . 3 |
7 | elxpi 4555 | . . . . . . 7 | |
8 | nqnq0pi 7246 | . . . . . . . . . . 11 ~Q0 | |
9 | 8 | adantl 275 | . . . . . . . . . 10 ~Q0 |
10 | eceq1 6464 | . . . . . . . . . . . 12 ~Q0 ~Q0 | |
11 | eceq1 6464 | . . . . . . . . . . . 12 | |
12 | 10, 11 | eqeq12d 2154 | . . . . . . . . . . 11 ~Q0 ~Q0 |
13 | 12 | adantr 274 | . . . . . . . . . 10 ~Q0 ~Q0 |
14 | 9, 13 | mpbird 166 | . . . . . . . . 9 ~Q0 |
15 | pinn 7117 | . . . . . . . . . . . . 13 | |
16 | opelxpi 4571 | . . . . . . . . . . . . 13 | |
17 | 15, 16 | sylan 281 | . . . . . . . . . . . 12 |
18 | 17 | adantl 275 | . . . . . . . . . . 11 |
19 | eleq1 2202 | . . . . . . . . . . . 12 | |
20 | 19 | adantr 274 | . . . . . . . . . . 11 |
21 | 18, 20 | mpbird 166 | . . . . . . . . . 10 |
22 | enq0ex 7247 | . . . . . . . . . . . 12 ~Q0 | |
23 | 22 | ecelqsi 6483 | . . . . . . . . . . 11 ~Q0 ~Q0 |
24 | df-nq0 7233 | . . . . . . . . . . 11 Q0 ~Q0 | |
25 | 23, 24 | eleqtrrdi 2233 | . . . . . . . . . 10 ~Q0 Q0 |
26 | 21, 25 | syl 14 | . . . . . . . . 9 ~Q0 Q0 |
27 | 14, 26 | eqeltrrd 2217 | . . . . . . . 8 Q0 |
28 | 27 | exlimivv 1868 | . . . . . . 7 Q0 |
29 | 7, 28 | syl 14 | . . . . . 6 Q0 |
30 | 29 | adantr 274 | . . . . 5 Q0 |
31 | eleq1 2202 | . . . . . 6 Q0 Q0 | |
32 | 31 | adantl 275 | . . . . 5 Q0 Q0 |
33 | 30, 32 | mpbird 166 | . . . 4 Q0 |
34 | 33 | exlimiv 1577 | . . 3 Q0 |
35 | 6, 34 | sylbi 120 | . 2 Q0 |
36 | 35 | ssriv 3101 | 1 Q0 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 wrex 2417 wss 3071 cop 3530 com 4504 cxp 4537 cec 6427 cqs 6428 cnpi 7080 ceq 7087 cnq 7088 ~Q0 ceq0 7094 Q0cnq0 7095 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-mi 7114 df-enq 7155 df-nqqs 7156 df-enq0 7232 df-nq0 7233 |
This theorem is referenced by: prarloclem5 7308 prarloclemcalc 7310 |
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