Description: Alternate proof of alephf1 . (Contributed by Mario Carneiro, 15-Mar-2013) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | alephf1ALT | |- aleph : On -1-1-> On |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | alephfnon | |- aleph Fn On |
|
| 2 | alephon | |- ( aleph ` x ) e. On |
|
| 3 | 2 | a1i | |- ( x e. On -> ( aleph ` x ) e. On ) |
| 4 | 3 | rgen | |- A. x e. On ( aleph ` x ) e. On |
| 5 | ffnfv | |- ( aleph : On --> On <-> ( aleph Fn On /\ A. x e. On ( aleph ` x ) e. On ) ) |
|
| 6 | 1 4 5 | mpbir2an | |- aleph : On --> On |
| 7 | alephsmo | |- Smo aleph |
|
| 8 | smo11 | |- ( ( aleph : On --> On /\ Smo aleph ) -> aleph : On -1-1-> On ) |
|
| 9 | 6 7 8 | mp2an | |- aleph : On -1-1-> On |