Description: Shorter proof of cnvfi using ax-pow . (Contributed by Mario Carneiro, 28-Dec-2014) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | cnvfiALT | |- ( A e. Fin -> `' A e. Fin ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnvcnvss | |- `' `' A C_ A |
|
| 2 | ssfi | |- ( ( A e. Fin /\ `' `' A C_ A ) -> `' `' A e. Fin ) |
|
| 3 | 1 2 | mpan2 | |- ( A e. Fin -> `' `' A e. Fin ) |
| 4 | relcnv | |- Rel `' A |
|
| 5 | cnvexg | |- ( A e. Fin -> `' A e. _V ) |
|
| 6 | cnven | |- ( ( Rel `' A /\ `' A e. _V ) -> `' A ~~ `' `' A ) |
|
| 7 | 4 5 6 | sylancr | |- ( A e. Fin -> `' A ~~ `' `' A ) |
| 8 | enfii | |- ( ( `' `' A e. Fin /\ `' A ~~ `' `' A ) -> `' A e. Fin ) |
|
| 9 | 3 7 8 | syl2anc | |- ( A e. Fin -> `' A e. Fin ) |