Description: A singleton is well-founded if its element is. (Contributed by Mario Carneiro, 10-Jun-2013) (Revised by Mario Carneiro, 16-Nov-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | snwf | |- ( A e. U. ( R1 " On ) -> { A } e. U. ( R1 " On ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwwf | |- ( A e. U. ( R1 " On ) <-> ~P A e. U. ( R1 " On ) ) |
|
2 | snsspw | |- { A } C_ ~P A |
|
3 | sswf | |- ( ( ~P A e. U. ( R1 " On ) /\ { A } C_ ~P A ) -> { A } e. U. ( R1 " On ) ) |
|
4 | 2 3 | mpan2 | |- ( ~P A e. U. ( R1 " On ) -> { A } e. U. ( R1 " On ) ) |
5 | 1 4 | sylbi | |- ( A e. U. ( R1 " On ) -> { A } e. U. ( R1 " On ) ) |