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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-enq0 7448 |
. . . . 5
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2 | 1 | relopabi 4767 |
. . . 4
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3 | 2 | a1i 9 |
. . 3
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4 | enq0sym 7456 |
. . . 4
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5 | 4 | adantl 277 |
. . 3
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6 | enq0tr 7458 |
. . . 4
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7 | 6 | adantl 277 |
. . 3
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8 | enq0ref 7457 |
. . . 4
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9 | 8 | a1i 9 |
. . 3
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10 | 3, 5, 7, 9 | iserd 6580 |
. 2
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11 | 10 | mptru 1373 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-recs 6325 df-irdg 6390 df-oadd 6440 df-omul 6441 df-er 6554 df-ni 7328 df-enq0 7448 |
This theorem is referenced by: enq0eceq 7461 nqnq0pi 7462 mulcanenq0ec 7469 nnnq0lem1 7470 addnq0mo 7471 mulnq0mo 7472 |
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