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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 7440 which says that a number apart from zero has a reciprocal). The defining characteristics of an apartness are irreflexivity (apirr 7389), symmetry (apsym 7390), and cotransitivity (apcotr 7391). Apartness implies negated equality, as seen at apne 7407, 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 7406). (Contributed by Jim Kingdon, 26-Jan-2020.) |
Ref | Expression |
---|---|
df-ap |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cap 7365 |
. 2
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2 | vx |
. . . . . . . . . . 11
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3 | 2 | cv 1241 |
. . . . . . . . . 10
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4 | vr |
. . . . . . . . . . . 12
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5 | 4 | cv 1241 |
. . . . . . . . . . 11
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6 | ci 6713 |
. . . . . . . . . . . 12
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7 | vs |
. . . . . . . . . . . . 13
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8 | 7 | cv 1241 |
. . . . . . . . . . . 12
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9 | cmul 6716 |
. . . . . . . . . . . 12
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10 | 6, 8, 9 | co 5455 |
. . . . . . . . . . 11
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11 | caddc 6714 |
. . . . . . . . . . 11
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12 | 5, 10, 11 | co 5455 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
13 | 3, 12 | wceq 1242 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | vy |
. . . . . . . . . . 11
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15 | 14 | cv 1241 |
. . . . . . . . . 10
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16 | vt |
. . . . . . . . . . . 12
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17 | 16 | cv 1241 |
. . . . . . . . . . 11
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18 | vu |
. . . . . . . . . . . . 13
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19 | 18 | cv 1241 |
. . . . . . . . . . . 12
![]() ![]() |
20 | 6, 19, 9 | co 5455 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() |
21 | 17, 20, 11 | co 5455 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 15, 21 | wceq 1242 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 13, 22 | wa 97 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | creap 7358 |
. . . . . . . . . 10
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25 | 5, 17, 24 | wbr 3755 |
. . . . . . . . 9
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26 | 8, 19, 24 | wbr 3755 |
. . . . . . . . 9
![]() ![]() ![]() |
27 | 25, 26 | wo 628 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 23, 27 | wa 97 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | cr 6710 |
. . . . . . 7
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30 | 28, 18, 29 | wrex 2301 |
. . . . . 6
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31 | 30, 16, 29 | wrex 2301 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
32 | 31, 7, 29 | wrex 2301 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 32, 4, 29 | wrex 2301 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
34 | 33, 2, 14 | copab 3808 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 1, 34 | wceq 1242 |
1
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Colors of variables: wff set class |
This definition is referenced by: apreap 7371 apreim 7387 |
Copyright terms: Public domain | W3C validator |