![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > df-er | Unicode version |
Description: Define the equivalence relation predicate. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 6043 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 6062, ersymb 6056, and ertr 6057. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.) |
Ref | Expression |
---|---|
df-er |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA |
. . 3
![]() ![]() | |
2 | cR |
. . 3
![]() ![]() | |
3 | 1, 2 | wer 6039 |
. 2
![]() ![]() ![]() ![]() |
4 | 2 | wrel 4293 |
. . 3
![]() ![]() ![]() |
5 | 2 | cdm 4288 |
. . . 4
![]() ![]() ![]() |
6 | 5, 1 | wceq 1242 |
. . 3
![]() ![]() ![]() ![]() ![]() |
7 | 2 | ccnv 4287 |
. . . . 5
![]() ![]() ![]() |
8 | 2, 2 | ccom 4292 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() |
9 | 7, 8 | cun 2909 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | 9, 2 | wss 2911 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
11 | 4, 6, 10 | w3a 884 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 3, 11 | wb 98 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
This definition is referenced by: dfer2 6043 ereq1 6049 ereq2 6050 errel 6051 erdm 6052 ersym 6054 ertr 6057 xpiderm 6113 |
Copyright terms: Public domain | W3C validator |