Metamath Proof Explorer

Theorem alephordilem1

Description: Lemma for alephordi . (Contributed by NM, 23-Oct-2009) (Revised by Mario Carneiro, 15-May-2015)

Ref Expression
Assertion alephordilem1 
|- ( A e. On -> ( aleph ` A ) ~< ( aleph ` suc A ) )

Proof

Step Hyp Ref Expression
1 alephon 
 |-  ( aleph ` A ) e. On
2 onenon 
 |-  ( ( aleph ` A ) e. On -> ( aleph ` A ) e. dom card )
3 harsdom 
 |-  ( ( aleph ` A ) e. dom card -> ( aleph ` A ) ~< ( har ` ( aleph ` A ) ) )
4 1 2 3 mp2b 
 |-  ( aleph ` A ) ~< ( har ` ( aleph ` A ) )
5 alephsuc 
 |-  ( A e. On -> ( aleph ` suc A ) = ( har ` ( aleph ` A ) ) )
6 4 5 breqtrrid 
 |-  ( A e. On -> ( aleph ` A ) ~< ( aleph ` suc A ) )