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Mirrors > Home > ILE Home > Th. List > nqnq0 | Unicode version |
Description: A positive fraction is a non-negative fraction. (Contributed by Jim Kingdon, 18-Nov-2019.) |
Ref | Expression |
---|---|
nqnq0 | Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nqqs 6538 | . . . . 5 | |
2 | 1 | eleq2i 2145 | . . . 4 |
3 | vex 2604 | . . . . 5 | |
4 | 3 | elqs 6180 | . . . 4 |
5 | df-rex 2354 | . . . 4 | |
6 | 2, 4, 5 | 3bitri 204 | . . 3 |
7 | elxpi 4379 | . . . . . . 7 | |
8 | nqnq0pi 6628 | . . . . . . . . . . 11 ~Q0 | |
9 | 8 | adantl 271 | . . . . . . . . . 10 ~Q0 |
10 | eceq1 6164 | . . . . . . . . . . . 12 ~Q0 ~Q0 | |
11 | eceq1 6164 | . . . . . . . . . . . 12 | |
12 | 10, 11 | eqeq12d 2095 | . . . . . . . . . . 11 ~Q0 ~Q0 |
13 | 12 | adantr 270 | . . . . . . . . . 10 ~Q0 ~Q0 |
14 | 9, 13 | mpbird 165 | . . . . . . . . 9 ~Q0 |
15 | pinn 6499 | . . . . . . . . . . . . 13 | |
16 | opelxpi 4394 | . . . . . . . . . . . . 13 | |
17 | 15, 16 | sylan 277 | . . . . . . . . . . . 12 |
18 | 17 | adantl 271 | . . . . . . . . . . 11 |
19 | eleq1 2141 | . . . . . . . . . . . 12 | |
20 | 19 | adantr 270 | . . . . . . . . . . 11 |
21 | 18, 20 | mpbird 165 | . . . . . . . . . 10 |
22 | enq0ex 6629 | . . . . . . . . . . . 12 ~Q0 | |
23 | 22 | ecelqsi 6183 | . . . . . . . . . . 11 ~Q0 ~Q0 |
24 | df-nq0 6615 | . . . . . . . . . . 11 Q0 ~Q0 | |
25 | 23, 24 | syl6eleqr 2172 | . . . . . . . . . 10 ~Q0 Q0 |
26 | 21, 25 | syl 14 | . . . . . . . . 9 ~Q0 Q0 |
27 | 14, 26 | eqeltrrd 2156 | . . . . . . . 8 Q0 |
28 | 27 | exlimivv 1817 | . . . . . . 7 Q0 |
29 | 7, 28 | syl 14 | . . . . . 6 Q0 |
30 | 29 | adantr 270 | . . . . 5 Q0 |
31 | eleq1 2141 | . . . . . 6 Q0 Q0 | |
32 | 31 | adantl 271 | . . . . 5 Q0 Q0 |
33 | 30, 32 | mpbird 165 | . . . 4 Q0 |
34 | 33 | exlimiv 1529 | . . 3 Q0 |
35 | 6, 34 | sylbi 119 | . 2 Q0 |
36 | 35 | ssriv 3003 | 1 Q0 |
Colors of variables: wff set class |
Syntax hints: wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 wrex 2349 wss 2973 cop 3401 com 4331 cxp 4361 cec 6127 cqs 6128 cnpi 6462 ceq 6469 cnq 6470 ~Q0 ceq0 6476 Q0cnq0 6477 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-mi 6496 df-enq 6537 df-nqqs 6538 df-enq0 6614 df-nq0 6615 |
This theorem is referenced by: prarloclem5 6690 prarloclemcalc 6692 |
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