Description: If R partially orders A , then the transitive predecessors are the same as the immediate predecessors . (Contributed by Scott Fenton, 28-Apr-2012) (Revised by Mario Carneiro, 26-Jun-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | trpredpo | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 | |
|
2 | simp3 | |
|
3 | predpo | |
|
4 | 3 | ralrimiv | |
5 | 4 | 3adant3 | |
6 | ssidd | |
|
7 | trpredmintr | |
|
8 | 1 2 5 6 7 | syl22anc | |
9 | setlikespec | |
|
10 | trpredpred | |
|
11 | 9 10 | syl | |
12 | 11 | 3adant1 | |
13 | 8 12 | eqssd | |