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Mirrors > Home > ILE Home > Th. List > enq0breq | Unicode version |
Description: Equivalence relation for nonnegative fractions in terms of natural numbers. (Contributed by NM, 27-Aug-1995.) |
Ref | Expression |
---|---|
enq0breq | ~Q0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 5848 | . . . . . 6 | |
2 | oveq12 5848 | . . . . . 6 | |
3 | 1, 2 | eqeqan12d 2180 | . . . . 5 |
4 | 3 | an42s 579 | . . . 4 |
5 | 4 | copsex4g 4222 | . . 3 |
6 | 5 | anbi2d 460 | . 2 |
7 | opexg 4203 | . . 3 | |
8 | opexg 4203 | . . 3 | |
9 | eleq1 2227 | . . . . . 6 | |
10 | 9 | anbi1d 461 | . . . . 5 |
11 | eqeq1 2171 | . . . . . . . 8 | |
12 | 11 | anbi1d 461 | . . . . . . 7 |
13 | 12 | anbi1d 461 | . . . . . 6 |
14 | 13 | 4exbidv 1857 | . . . . 5 |
15 | 10, 14 | anbi12d 465 | . . . 4 |
16 | eleq1 2227 | . . . . . 6 | |
17 | 16 | anbi2d 460 | . . . . 5 |
18 | eqeq1 2171 | . . . . . . . 8 | |
19 | 18 | anbi2d 460 | . . . . . . 7 |
20 | 19 | anbi1d 461 | . . . . . 6 |
21 | 20 | 4exbidv 1857 | . . . . 5 |
22 | 17, 21 | anbi12d 465 | . . . 4 |
23 | df-enq0 7359 | . . . 4 ~Q0 | |
24 | 15, 22, 23 | brabg 4244 | . . 3 ~Q0 |
25 | 7, 8, 24 | syl2an 287 | . 2 ~Q0 |
26 | opelxpi 4633 | . . . 4 | |
27 | opelxpi 4633 | . . . 4 | |
28 | 26, 27 | anim12i 336 | . . 3 |
29 | 28 | biantrurd 303 | . 2 |
30 | 6, 25, 29 | 3bitr4d 219 | 1 ~Q0 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 cvv 2724 cop 3576 class class class wbr 3979 com 4564 cxp 4599 (class class class)co 5839 comu 6376 cnpi 7207 ~Q0 ceq0 7221 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-pow 4150 ax-pr 4184 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2726 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-br 3980 df-opab 4041 df-xp 4607 df-iota 5150 df-fv 5193 df-ov 5842 df-enq0 7359 |
This theorem is referenced by: enq0eceq 7372 nqnq0pi 7373 addcmpblnq0 7378 mulcmpblnq0 7379 mulcanenq0ec 7380 nnnq0lem1 7381 |
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