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 ) )