Metamath Proof Explorer


Theorem domtri

Description: Trichotomy law for dominance and strict dominance. This theorem is equivalent to the Axiom of Choice. (Contributed by NM, 4-Jan-2004) (Revised by Mario Carneiro, 30-Apr-2015)

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

Proof

Step Hyp Ref Expression
1 numth3
 |-  ( A e. V -> A e. dom card )
2 numth3
 |-  ( B e. W -> B e. dom card )
3 domtri2
 |-  ( ( A e. dom card /\ B e. dom card ) -> ( A ~<_ B <-> -. B ~< A ) )
4 1 2 3 syl2an
 |-  ( ( A e. V /\ B e. W ) -> ( A ~<_ B <-> -. B ~< A ) )