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Mirrors > Home > ILE Home > Th. List > enq0er | Unicode version |
Description: The equivalence relation for nonnegative fractions is an equivalence relation. (Contributed by Jim Kingdon, 12-Nov-2019.) |
Ref | Expression |
---|---|
enq0er | ~Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-enq0 7225 | . . . . 5 ~Q0 | |
2 | 1 | relopabi 4660 | . . . 4 ~Q0 |
3 | 2 | a1i 9 | . . 3 ~Q0 |
4 | enq0sym 7233 | . . . 4 ~Q0 ~Q0 | |
5 | 4 | adantl 275 | . . 3 ~Q0 ~Q0 |
6 | enq0tr 7235 | . . . 4 ~Q0 ~Q0 ~Q0 | |
7 | 6 | adantl 275 | . . 3 ~Q0 ~Q0 ~Q0 |
8 | enq0ref 7234 | . . . 4 ~Q0 | |
9 | 8 | a1i 9 | . . 3 ~Q0 |
10 | 3, 5, 7, 9 | iserd 6448 | . 2 ~Q0 |
11 | 10 | mptru 1340 | 1 ~Q0 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1331 wtru 1332 wex 1468 wcel 1480 cop 3525 class class class wbr 3924 com 4499 cxp 4532 wrel 4539 (class class class)co 5767 comu 6304 wer 6419 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-dc 820 df-3or 963 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-er 6422 df-ni 7105 df-enq0 7225 |
This theorem is referenced by: enq0eceq 7238 nqnq0pi 7239 mulcanenq0ec 7246 nnnq0lem1 7247 addnq0mo 7248 mulnq0mo 7249 |
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