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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 7236. (Contributed by Jim Kingdon, 14-Nov-2019.) |
Ref | Expression |
---|---|
enq0ref | ~Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxpi 4550 | . . . . . 6 | |
2 | elxpi 4550 | . . . . . 6 | |
3 | ee4anv 1904 | . . . . . 6 | |
4 | 1, 2, 3 | sylanbrc 413 | . . . . 5 |
5 | eqtr2 2156 | . . . . . . . . . . . 12 | |
6 | vex 2684 | . . . . . . . . . . . . 13 | |
7 | vex 2684 | . . . . . . . . . . . . 13 | |
8 | 6, 7 | opth 4154 | . . . . . . . . . . . 12 |
9 | 5, 8 | sylib 121 | . . . . . . . . . . 11 |
10 | oveq1 5774 | . . . . . . . . . . . 12 | |
11 | oveq2 5775 | . . . . . . . . . . . . 13 | |
12 | 11 | equcoms 1684 | . . . . . . . . . . . 12 |
13 | 10, 12 | sylan9eq 2190 | . . . . . . . . . . 11 |
14 | 9, 13 | syl 14 | . . . . . . . . . 10 |
15 | 14 | ancli 321 | . . . . . . . . 9 |
16 | 15 | ad2ant2r 500 | . . . . . . . 8 |
17 | pinn 7110 | . . . . . . . . . . . . . 14 | |
18 | nnmcom 6378 | . . . . . . . . . . . . . 14 | |
19 | 17, 18 | sylan2 284 | . . . . . . . . . . . . 13 |
20 | 19 | eqeq2d 2149 | . . . . . . . . . . . 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 2684 | . . . . 5 | |
31 | eleq1 2200 | . . . . . . 7 | |
32 | 31 | anbi1d 460 | . . . . . 6 |
33 | eqeq1 2144 | . . . . . . . . 9 | |
34 | 33 | anbi1d 460 | . . . . . . . 8 |
35 | 34 | anbi1d 460 | . . . . . . 7 |
36 | 35 | 4exbidv 1842 | . . . . . 6 |
37 | 32, 36 | anbi12d 464 | . . . . 5 |
38 | eleq1 2200 | . . . . . . 7 | |
39 | 38 | anbi2d 459 | . . . . . 6 |
40 | eqeq1 2144 | . . . . . . . . 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 7225 | . . . . 5 ~Q0 | |
46 | 30, 30, 37, 44, 45 | brab 4189 | . . . 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 3525 class class class wbr 3924 com 4499 cxp 4532 (class class class)co 5767 comu 6304 cnpi 7073 ~Q0 ceq0 7087 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-oadd 6310 df-omul 6311 df-ni 7105 df-enq0 7225 |
This theorem is referenced by: enq0er 7236 |
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