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Metamath Proof Explorer

Theorem alephfp2

Description: The aleph function has at least one fixed point. Proposition 11.18 of TakeutiZaring p. 104. See alephfp for an actual example of a fixed point. Compare the inequality alephle that holds in general. Note that if x is a fixed point, then alephalephaleph` ... aleph `x = x . (Contributed by NM, 6-Nov-2004) (Revised by Mario Carneiro, 15-May-2015)

Ref Expression
Assertion alephfp2 
|- E. x e. On ( aleph ` x ) = x

Proof

Step Hyp Ref Expression
1 alephsson 
 |-  ran aleph C_ On
2 eqid 
 |-  ( rec ( aleph , _om ) |` _om ) = ( rec ( aleph , _om ) |` _om )
3 2 alephfplem4 
 |-  U. ( ( rec ( aleph , _om ) |` _om ) " _om ) e. ran aleph
4 1 3 sselii 
 |-  U. ( ( rec ( aleph , _om ) |` _om ) " _om ) e. On
5 2 alephfp 
 |-  ( aleph ` U. ( ( rec ( aleph , _om ) |` _om ) " _om ) ) = U. ( ( rec ( aleph , _om ) |` _om ) " _om )
6 fveq2 
 |-  ( x = U. ( ( rec ( aleph , _om ) |` _om ) " _om ) -> ( aleph ` x ) = ( aleph ` U. ( ( rec ( aleph , _om ) |` _om ) " _om ) ) )
7 id 
 |-  ( x = U. ( ( rec ( aleph , _om ) |` _om ) " _om ) -> x = U. ( ( rec ( aleph , _om ) |` _om ) " _om ) )
8 6 7 eqeq12d 
 |-  ( x = U. ( ( rec ( aleph , _om ) |` _om ) " _om ) -> ( ( aleph ` x ) = x <-> ( aleph ` U. ( ( rec ( aleph , _om ) |` _om ) " _om ) ) = U. ( ( rec ( aleph , _om ) |` _om ) " _om ) ) )
9 8 rspcev 
 |-  ( ( U. ( ( rec ( aleph , _om ) |` _om ) " _om ) e. On /\ ( aleph ` U. ( ( rec ( aleph , _om ) |` _om ) " _om ) ) = U. ( ( rec ( aleph , _om ) |` _om ) " _om ) ) -> E. x e. On ( aleph ` x ) = x )
10 4 5 9 mp2an 
 |-  E. x e. On ( aleph ` x ) = x