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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 7310 | . . . . 5 | |
2 | 1 | eleq2i 2237 | . . . 4 |
3 | vex 2733 | . . . . 5 | |
4 | 3 | elqs 6564 | . . . 4 |
5 | df-rex 2454 | . . . 4 | |
6 | 2, 4, 5 | 3bitri 205 | . . 3 |
7 | elxpi 4627 | . . . . . . 7 | |
8 | nqnq0pi 7400 | . . . . . . . . . . 11 ~Q0 | |
9 | 8 | adantl 275 | . . . . . . . . . 10 ~Q0 |
10 | eceq1 6548 | . . . . . . . . . . . 12 ~Q0 ~Q0 | |
11 | eceq1 6548 | . . . . . . . . . . . 12 | |
12 | 10, 11 | eqeq12d 2185 | . . . . . . . . . . 11 ~Q0 ~Q0 |
13 | 12 | adantr 274 | . . . . . . . . . 10 ~Q0 ~Q0 |
14 | 9, 13 | mpbird 166 | . . . . . . . . 9 ~Q0 |
15 | pinn 7271 | . . . . . . . . . . . . 13 | |
16 | opelxpi 4643 | . . . . . . . . . . . . 13 | |
17 | 15, 16 | sylan 281 | . . . . . . . . . . . 12 |
18 | 17 | adantl 275 | . . . . . . . . . . 11 |
19 | eleq1 2233 | . . . . . . . . . . . 12 | |
20 | 19 | adantr 274 | . . . . . . . . . . 11 |
21 | 18, 20 | mpbird 166 | . . . . . . . . . 10 |
22 | enq0ex 7401 | . . . . . . . . . . . 12 ~Q0 | |
23 | 22 | ecelqsi 6567 | . . . . . . . . . . 11 ~Q0 ~Q0 |
24 | df-nq0 7387 | . . . . . . . . . . 11 Q0 ~Q0 | |
25 | 23, 24 | eleqtrrdi 2264 | . . . . . . . . . 10 ~Q0 Q0 |
26 | 21, 25 | syl 14 | . . . . . . . . 9 ~Q0 Q0 |
27 | 14, 26 | eqeltrrd 2248 | . . . . . . . 8 Q0 |
28 | 27 | exlimivv 1889 | . . . . . . 7 Q0 |
29 | 7, 28 | syl 14 | . . . . . 6 Q0 |
30 | 29 | adantr 274 | . . . . 5 Q0 |
31 | eleq1 2233 | . . . . . 6 Q0 Q0 | |
32 | 31 | adantl 275 | . . . . 5 Q0 Q0 |
33 | 30, 32 | mpbird 166 | . . . 4 Q0 |
34 | 33 | exlimiv 1591 | . . 3 Q0 |
35 | 6, 34 | sylbi 120 | . 2 Q0 |
36 | 35 | ssriv 3151 | 1 Q0 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1348 wex 1485 wcel 2141 wrex 2449 wss 3121 cop 3586 com 4574 cxp 4609 cec 6511 cqs 6512 cnpi 7234 ceq 7241 cnq 7242 ~Q0 ceq0 7248 Q0cnq0 7249 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-mi 7268 df-enq 7309 df-nqqs 7310 df-enq0 7386 df-nq0 7387 |
This theorem is referenced by: prarloclem5 7462 prarloclemcalc 7464 |
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