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Mirrors > Home > ILE Home > Th. List > enq0ref | Unicode version |
Description: The equivalence relation for nonnegative fractions is reflexive. Lemma for enq0er 7243. (Contributed by Jim Kingdon, 14-Nov-2019.) |
Ref | Expression |
---|---|
enq0ref | ~Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxpi 4555 | . . . . . 6 | |
2 | elxpi 4555 | . . . . . 6 | |
3 | ee4anv 1906 | . . . . . 6 | |
4 | 1, 2, 3 | sylanbrc 413 | . . . . 5 |
5 | eqtr2 2158 | . . . . . . . . . . . 12 | |
6 | vex 2689 | . . . . . . . . . . . . 13 | |
7 | vex 2689 | . . . . . . . . . . . . 13 | |
8 | 6, 7 | opth 4159 | . . . . . . . . . . . 12 |
9 | 5, 8 | sylib 121 | . . . . . . . . . . 11 |
10 | oveq1 5781 | . . . . . . . . . . . 12 | |
11 | oveq2 5782 | . . . . . . . . . . . . 13 | |
12 | 11 | equcoms 1684 | . . . . . . . . . . . 12 |
13 | 10, 12 | sylan9eq 2192 | . . . . . . . . . . 11 |
14 | 9, 13 | syl 14 | . . . . . . . . . 10 |
15 | 14 | ancli 321 | . . . . . . . . 9 |
16 | 15 | ad2ant2r 500 | . . . . . . . 8 |
17 | pinn 7117 | . . . . . . . . . . . . . 14 | |
18 | nnmcom 6385 | . . . . . . . . . . . . . 14 | |
19 | 17, 18 | sylan2 284 | . . . . . . . . . . . . 13 |
20 | 19 | eqeq2d 2151 | . . . . . . . . . . . 12 |
21 | 20 | ancoms 266 | . . . . . . . . . . 11 |
22 | 21 | ad2ant2lr 501 | . . . . . . . . . 10 |
23 | 22 | ad2ant2l 499 | . . . . . . . . 9 |
24 | 23 | anbi2d 459 | . . . . . . . 8 |
25 | 16, 24 | mpbid 146 | . . . . . . 7 |
26 | 25 | 2eximi 1580 | . . . . . 6 |
27 | 26 | 2eximi 1580 | . . . . 5 |
28 | 4, 27 | syl 14 | . . . 4 |
29 | 28 | ancli 321 | . . 3 |
30 | vex 2689 | . . . . 5 | |
31 | eleq1 2202 | . . . . . . 7 | |
32 | 31 | anbi1d 460 | . . . . . 6 |
33 | eqeq1 2146 | . . . . . . . . 9 | |
34 | 33 | anbi1d 460 | . . . . . . . 8 |
35 | 34 | anbi1d 460 | . . . . . . 7 |
36 | 35 | 4exbidv 1842 | . . . . . 6 |
37 | 32, 36 | anbi12d 464 | . . . . 5 |
38 | eleq1 2202 | . . . . . . 7 | |
39 | 38 | anbi2d 459 | . . . . . 6 |
40 | eqeq1 2146 | . . . . . . . . 9 | |
41 | 40 | anbi2d 459 | . . . . . . . 8 |
42 | 41 | anbi1d 460 | . . . . . . 7 |
43 | 42 | 4exbidv 1842 | . . . . . 6 |
44 | 39, 43 | anbi12d 464 | . . . . 5 |
45 | df-enq0 7232 | . . . . 5 ~Q0 | |
46 | 30, 30, 37, 44, 45 | brab 4194 | . . . 4 ~Q0 |
47 | anidm 393 | . . . . 5 | |
48 | 47 | anbi1i 453 | . . . 4 |
49 | 46, 48 | bitri 183 | . . 3 ~Q0 |
50 | 29, 49 | sylibr 133 | . 2 ~Q0 |
51 | 49 | simplbi 272 | . 2 ~Q0 |
52 | 50, 51 | impbii 125 | 1 ~Q0 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cop 3530 class class class wbr 3929 com 4504 cxp 4537 (class class class)co 5774 comu 6311 cnpi 7080 ~Q0 ceq0 7094 |
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-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-ni 7112 df-enq0 7232 |
This theorem is referenced by: enq0er 7243 |
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