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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 5783 | . . . . . 6 | |
2 | oveq12 5783 | . . . . . 6 | |
3 | 1, 2 | eqeqan12d 2155 | . . . . 5 |
4 | 3 | an42s 578 | . . . 4 |
5 | 4 | copsex4g 4169 | . . 3 |
6 | 5 | anbi2d 459 | . 2 |
7 | opexg 4150 | . . 3 | |
8 | opexg 4150 | . . 3 | |
9 | eleq1 2202 | . . . . . 6 | |
10 | 9 | anbi1d 460 | . . . . 5 |
11 | eqeq1 2146 | . . . . . . . 8 | |
12 | 11 | anbi1d 460 | . . . . . . 7 |
13 | 12 | anbi1d 460 | . . . . . 6 |
14 | 13 | 4exbidv 1842 | . . . . 5 |
15 | 10, 14 | anbi12d 464 | . . . 4 |
16 | eleq1 2202 | . . . . . 6 | |
17 | 16 | anbi2d 459 | . . . . 5 |
18 | eqeq1 2146 | . . . . . . . 8 | |
19 | 18 | anbi2d 459 | . . . . . . 7 |
20 | 19 | anbi1d 460 | . . . . . 6 |
21 | 20 | 4exbidv 1842 | . . . . 5 |
22 | 17, 21 | anbi12d 464 | . . . 4 |
23 | df-enq0 7232 | . . . 4 ~Q0 | |
24 | 15, 22, 23 | brabg 4191 | . . 3 ~Q0 |
25 | 7, 8, 24 | syl2an 287 | . 2 ~Q0 |
26 | opelxpi 4571 | . . . 4 | |
27 | opelxpi 4571 | . . . 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 1331 wex 1468 wcel 1480 cvv 2686 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-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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 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-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-iota 5088 df-fv 5131 df-ov 5777 df-enq0 7232 |
This theorem is referenced by: enq0eceq 7245 nqnq0pi 7246 addcmpblnq0 7251 mulcmpblnq0 7252 mulcanenq0ec 7253 nnnq0lem1 7254 |
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