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Mirrors > Home > ILE Home > Th. List > df-ap | Unicode version |
Description: Define complex apartness.
Definition 6.1 of [Geuvers], p. 17.
Two numbers are considered apart if it is possible to separate them. One common usage is that we can divide by a number if it is apart from zero (see for example recclap 8463 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 8391), symmetry (apsym 8392), and cotransitivity (apcotr 8393). Apartness implies negated equality, as seen at apne 8409, and the converse would also follow if we assumed excluded middle. In addition, apartness of complex numbers is tight, which means that two numbers which are not apart are equal (apti 8408). (Contributed by Jim Kingdon, 26-Jan-2020.) |
Ref | Expression |
---|---|
df-ap |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cap 8367 |
. 2
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2 | vx |
. . . . . . . . . . 11
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3 | 2 | cv 1331 |
. . . . . . . . . 10
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4 | vr |
. . . . . . . . . . . 12
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5 | 4 | cv 1331 |
. . . . . . . . . . 11
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6 | ci 7646 |
. . . . . . . . . . . 12
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7 | vs |
. . . . . . . . . . . . 13
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8 | 7 | cv 1331 |
. . . . . . . . . . . 12
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9 | cmul 7649 |
. . . . . . . . . . . 12
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10 | 6, 8, 9 | co 5782 |
. . . . . . . . . . 11
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11 | caddc 7647 |
. . . . . . . . . . 11
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12 | 5, 10, 11 | co 5782 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 3, 12 | wceq 1332 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | vy |
. . . . . . . . . . 11
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15 | 14 | cv 1331 |
. . . . . . . . . 10
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16 | vt |
. . . . . . . . . . . 12
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17 | 16 | cv 1331 |
. . . . . . . . . . 11
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18 | vu |
. . . . . . . . . . . . 13
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19 | 18 | cv 1331 |
. . . . . . . . . . . 12
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20 | 6, 19, 9 | co 5782 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() |
21 | 17, 20, 11 | co 5782 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 15, 21 | wceq 1332 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 13, 22 | wa 103 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | creap 8360 |
. . . . . . . . . 10
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25 | 5, 17, 24 | wbr 3937 |
. . . . . . . . 9
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26 | 8, 19, 24 | wbr 3937 |
. . . . . . . . 9
![]() ![]() ![]() |
27 | 25, 26 | wo 698 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 23, 27 | wa 103 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | cr 7643 |
. . . . . . 7
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30 | 28, 18, 29 | wrex 2418 |
. . . . . 6
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31 | 30, 16, 29 | wrex 2418 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 31, 7, 29 | wrex 2418 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 32, 4, 29 | wrex 2418 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 33, 2, 14 | copab 3996 |
. 2
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35 | 1, 34 | wceq 1332 |
1
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Colors of variables: wff set class |
This definition is referenced by: apreap 8373 apreim 8389 aprcl 8432 |
Copyright terms: Public domain | W3C validator |