Metamath Proof Explorer

Theorem cardsdom2

Description: A numerable set is strictly dominated by another iff their cardinalities are strictly ordered. (Contributed by Stefan O'Rear, 30-Oct-2014) (Revised by Mario Carneiro, 29-Apr-2015)

Ref Expression
Assertion cardsdom2 
|- ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) e. ( card ` B ) <-> A ~< B ) )

Proof

Step Hyp Ref Expression
1 carddom2 
 |-  ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) C_ ( card ` B ) <-> A ~<_ B ) )
2 carden2 
 |-  ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) = ( card ` B ) <-> A ~~ B ) )
3 2 necon3abid 
 |-  ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) =/= ( card ` B ) <-> -. A ~~ B ) )
4 1 3 anbi12d 
 |-  ( ( A e. dom card /\ B e. dom card ) -> ( ( ( card ` A ) C_ ( card ` B ) /\ ( card ` A ) =/= ( card ` B ) ) <-> ( A ~<_ B /\ -. A ~~ B ) ) )
5 cardon 
 |-  ( card ` A ) e. On
6 cardon 
 |-  ( card ` B ) e. On
7 onelpss 
 |-  ( ( ( card ` A ) e. On /\ ( card ` B ) e. On ) -> ( ( card ` A ) e. ( card ` B ) <-> ( ( card ` A ) C_ ( card ` B ) /\ ( card ` A ) =/= ( card ` B ) ) ) )
8 5 6 7 mp2an 
 |-  ( ( card ` A ) e. ( card ` B ) <-> ( ( card ` A ) C_ ( card ` B ) /\ ( card ` A ) =/= ( card ` B ) ) )
9 brsdom 
 |-  ( A ~< B <-> ( A ~<_ B /\ -. A ~~ B ) )
10 4 8 9 3bitr4g 
 |-  ( ( A e. dom card /\ B e. dom card ) -> ( ( card ` A ) e. ( card ` B ) <-> A ~< B ) )