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Metamath Proof Explorer

Theorem fidomtri

Description: Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014) (Revised by Mario Carneiro, 27-Apr-2015)

Ref Expression
Assertion fidomtri 
|- ( ( A e. Fin /\ B e. V ) -> ( A ~<_ B <-> -. B ~< A ) )

Proof

Step Hyp Ref Expression
1 domnsym 
 |-  ( A ~<_ B -> -. B ~< A )
2 finnum 
 |-  ( A e. Fin -> A e. dom card )
3 2 adantr 
 |-  ( ( A e. Fin /\ B e. V ) -> A e. dom card )
4 finnum 
 |-  ( B e. Fin -> B e. dom card )
5 domtri2 
 |-  ( ( A e. dom card /\ B e. dom card ) -> ( A ~<_ B <-> -. B ~< A ) )
6 3 4 5 syl2an 
 |-  ( ( ( A e. Fin /\ B e. V ) /\ B e. Fin ) -> ( A ~<_ B <-> -. B ~< A ) )
7 6 biimprd 
 |-  ( ( ( A e. Fin /\ B e. V ) /\ B e. Fin ) -> ( -. B ~< A -> A ~<_ B ) )
8 isinffi 
 |-  ( ( -. B e. Fin /\ A e. Fin ) -> E. a a : A -1-1-> B )
9 8 ancoms 
 |-  ( ( A e. Fin /\ -. B e. Fin ) -> E. a a : A -1-1-> B )
10 9 adantlr 
 |-  ( ( ( A e. Fin /\ B e. V ) /\ -. B e. Fin ) -> E. a a : A -1-1-> B )
11 brdomg 
 |-  ( B e. V -> ( A ~<_ B <-> E. a a : A -1-1-> B ) )
12 11 ad2antlr 
 |-  ( ( ( A e. Fin /\ B e. V ) /\ -. B e. Fin ) -> ( A ~<_ B <-> E. a a : A -1-1-> B ) )
13 10 12 mpbird 
 |-  ( ( ( A e. Fin /\ B e. V ) /\ -. B e. Fin ) -> A ~<_ B )
14 13 a1d 
 |-  ( ( ( A e. Fin /\ B e. V ) /\ -. B e. Fin ) -> ( -. B ~< A -> A ~<_ B ) )
15 7 14 pm2.61dan 
 |-  ( ( A e. Fin /\ B e. V ) -> ( -. B ~< A -> A ~<_ B ) )
16 1 15 impbid2 
 |-  ( ( A e. Fin /\ B e. V ) -> ( A ~<_ B <-> -. B ~< A ) )