Description: The cardinality of the successor of an infinite ordinal. (Contributed by Mario Carneiro, 11-Jan-2013)
Ref | Expression | ||
---|---|---|---|
Assertion | cardsucinf | |- ( ( A e. On /\ _om C_ A ) -> ( card ` suc A ) = ( card ` A ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infensuc | |- ( ( A e. On /\ _om C_ A ) -> A ~~ suc A ) |
|
2 | carden2b | |- ( A ~~ suc A -> ( card ` A ) = ( card ` suc A ) ) |
|
3 | 1 2 | syl | |- ( ( A e. On /\ _om C_ A ) -> ( card ` A ) = ( card ` suc A ) ) |
4 | 3 | eqcomd | |- ( ( A e. On /\ _om C_ A ) -> ( card ` suc A ) = ( card ` A ) ) |