Metamath Proof Explorer


Theorem alephdom

Description: Relationship between inclusion of ordinal numbers and dominance of infinite initial ordinals. (Contributed by Jeff Hankins, 23-Oct-2009)

Ref Expression
Assertion alephdom
|- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> ( aleph ` A ) ~<_ ( aleph ` B ) ) )

Proof

Step Hyp Ref Expression
1 onsseleq
 |-  ( ( A e. On /\ B e. On ) -> ( A C_ B <-> ( A e. B \/ A = B ) ) )
2 alephord
 |-  ( ( A e. On /\ B e. On ) -> ( A e. B <-> ( aleph ` A ) ~< ( aleph ` B ) ) )
3 sdomdom
 |-  ( ( aleph ` A ) ~< ( aleph ` B ) -> ( aleph ` A ) ~<_ ( aleph ` B ) )
4 2 3 syl6bi
 |-  ( ( A e. On /\ B e. On ) -> ( A e. B -> ( aleph ` A ) ~<_ ( aleph ` B ) ) )
5 fvex
 |-  ( aleph ` A ) e. _V
6 fveq2
 |-  ( A = B -> ( aleph ` A ) = ( aleph ` B ) )
7 eqeng
 |-  ( ( aleph ` A ) e. _V -> ( ( aleph ` A ) = ( aleph ` B ) -> ( aleph ` A ) ~~ ( aleph ` B ) ) )
8 5 6 7 mpsyl
 |-  ( A = B -> ( aleph ` A ) ~~ ( aleph ` B ) )
9 8 a1i
 |-  ( ( A e. On /\ B e. On ) -> ( A = B -> ( aleph ` A ) ~~ ( aleph ` B ) ) )
10 endom
 |-  ( ( aleph ` A ) ~~ ( aleph ` B ) -> ( aleph ` A ) ~<_ ( aleph ` B ) )
11 9 10 syl6
 |-  ( ( A e. On /\ B e. On ) -> ( A = B -> ( aleph ` A ) ~<_ ( aleph ` B ) ) )
12 4 11 jaod
 |-  ( ( A e. On /\ B e. On ) -> ( ( A e. B \/ A = B ) -> ( aleph ` A ) ~<_ ( aleph ` B ) ) )
13 1 12 sylbid
 |-  ( ( A e. On /\ B e. On ) -> ( A C_ B -> ( aleph ` A ) ~<_ ( aleph ` B ) ) )
14 eloni
 |-  ( B e. On -> Ord B )
15 eloni
 |-  ( A e. On -> Ord A )
16 ordtri2or
 |-  ( ( Ord B /\ Ord A ) -> ( B e. A \/ A C_ B ) )
17 14 15 16 syl2anr
 |-  ( ( A e. On /\ B e. On ) -> ( B e. A \/ A C_ B ) )
18 17 ord
 |-  ( ( A e. On /\ B e. On ) -> ( -. B e. A -> A C_ B ) )
19 18 con1d
 |-  ( ( A e. On /\ B e. On ) -> ( -. A C_ B -> B e. A ) )
20 alephord
 |-  ( ( B e. On /\ A e. On ) -> ( B e. A <-> ( aleph ` B ) ~< ( aleph ` A ) ) )
21 20 ancoms
 |-  ( ( A e. On /\ B e. On ) -> ( B e. A <-> ( aleph ` B ) ~< ( aleph ` A ) ) )
22 sdomnen
 |-  ( ( aleph ` B ) ~< ( aleph ` A ) -> -. ( aleph ` B ) ~~ ( aleph ` A ) )
23 sdomdom
 |-  ( ( aleph ` B ) ~< ( aleph ` A ) -> ( aleph ` B ) ~<_ ( aleph ` A ) )
24 sbth
 |-  ( ( ( aleph ` B ) ~<_ ( aleph ` A ) /\ ( aleph ` A ) ~<_ ( aleph ` B ) ) -> ( aleph ` B ) ~~ ( aleph ` A ) )
25 24 ex
 |-  ( ( aleph ` B ) ~<_ ( aleph ` A ) -> ( ( aleph ` A ) ~<_ ( aleph ` B ) -> ( aleph ` B ) ~~ ( aleph ` A ) ) )
26 23 25 syl
 |-  ( ( aleph ` B ) ~< ( aleph ` A ) -> ( ( aleph ` A ) ~<_ ( aleph ` B ) -> ( aleph ` B ) ~~ ( aleph ` A ) ) )
27 22 26 mtod
 |-  ( ( aleph ` B ) ~< ( aleph ` A ) -> -. ( aleph ` A ) ~<_ ( aleph ` B ) )
28 21 27 syl6bi
 |-  ( ( A e. On /\ B e. On ) -> ( B e. A -> -. ( aleph ` A ) ~<_ ( aleph ` B ) ) )
29 19 28 syld
 |-  ( ( A e. On /\ B e. On ) -> ( -. A C_ B -> -. ( aleph ` A ) ~<_ ( aleph ` B ) ) )
30 13 29 impcon4bid
 |-  ( ( A e. On /\ B e. On ) -> ( A C_ B <-> ( aleph ` A ) ~<_ ( aleph ` B ) ) )