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 ) ) |
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 ) ) |