Theorem entric

Description: Trichotomy of equinumerosity and strict dominance. This theorem is equivalent to the Axiom of Choice. Theorem 8 of Suppes p. 242. (Contributed by NM, 4-Jan-2004)

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

Proof

Step	Hyp	Ref	Expression
1		domtri	|- ( ( A e. V /\ B e. W ) -> ( A ~<_ B <-> -. B ~< A ) )
2	1	biimprd	|- ( ( A e. V /\ B e. W ) -> ( -. B ~< A -> A ~<_ B ) )
3		brdom2	|- ( A ~<_ B <-> ( A ~< B \/ A ~~ B ) )
4	2 3	imbitrdi	|- ( ( A e. V /\ B e. W ) -> ( -. B ~< A -> ( A ~< B \/ A ~~ B ) ) )
5	4	con1d	|- ( ( A e. V /\ B e. W ) -> ( -. ( A ~< B \/ A ~~ B ) -> B ~< A ) )
6	5	orrd	|- ( ( A e. V /\ B e. W ) -> ( ( A ~< B \/ A ~~ B ) \/ B ~< A ) )
7		df-3or	|- ( ( A ~< B \/ A ~~ B \/ B ~< A ) <-> ( ( A ~< B \/ A ~~ B ) \/ B ~< A ) )
8	6 7	sylibr	|- ( ( A e. V /\ B e. W ) -> ( A ~< B \/ A ~~ B \/ B ~< A ) )